Sine-Gordon equation

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class="vector-toc-link" href="#Frenkel–Kontorova_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Frenkel–Kontorova model</span> </div> </a> <ul id="toc-Frenkel–Kontorova_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_mechanical_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_mechanical_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>A mechanical model</span> </div> </a> <ul id="toc-A_mechanical_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Naming" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Naming"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Naming</span> </div> </a> <ul id="toc-Naming-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Soliton_solutions" class="vector-toc-list-item vector-toc-level-1 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class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>2-soliton solutions</span> </div> </a> <ul id="toc-2-soliton_solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-3-soliton_solutions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#3-soliton_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>3-soliton solutions</span> </div> </a> <ul id="toc-3-soliton_solutions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bäcklund_transformation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bäcklund_transformation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bäcklund transformation</span> </div> </a> <ul id="toc-Bäcklund_transformation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_charge_and_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topological_charge_and_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Topological charge and energy</span> </div> </a> <ul id="toc-Topological_charge_and_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zero-curvature_formulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zero-curvature_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Zero-curvature formulation</span> </div> </a> <ul id="toc-Zero-curvature_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Related equations</span> </div> </a> <ul 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<span>Toggle Quantum sine-Gordon model subsection</span> </button> <ul id="toc-Quantum_sine-Gordon_model-sublist" class="vector-toc-list"> <li id="toc-Regimes_of_renormalizability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regimes_of_renormalizability"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Regimes of renormalizability</span> </div> </a> <ul id="toc-Regimes_of_renormalizability-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stochastic_sine-Gordon_model" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stochastic_sine-Gordon_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Stochastic sine-Gordon model</span> </div> </a> <ul id="toc-Stochastic_sine-Gordon_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Supersymmetric_sine-Gordon_model" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Supersymmetric_sine-Gordon_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Supersymmetric sine-Gordon model</span> </div> </a> <ul id="toc-Supersymmetric_sine-Gordon_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physical_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Physical_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Physical applications</span> </div> </a> <ul id="toc-Physical_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown 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class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%BE%E0%A8%88%E0%A8%A8-%E0%A8%9C%E0%A9%8C%E0%A8%B0%E0%A8%A1%E0%A8%A8_%E0%A8%B8%E0%A8%AE%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8" title="ਸਾਈਨ-ਜੌਰਡਨ ਸਮੀਕਰਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਾਈਨ-ਜੌਰਡਨ ਸਮੀਕਰਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D1%81%D0%B8%D0%BD%D1%83%D1%81-%D0%93%D0%BE%D1%80%D0%B4%D0%BE%D0%BD%D0%B0" title="Уравнение синус-Гордона – Russian" lang="ru" hreflang="ru" data-title="Уравнение синус-Гордона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li 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searchaux" style="display:none">Nonlinear partial differential equation</div> <p>The <b>sine-Gordon equation</b> is a second-order <a href="/wiki/Nonlinear_partial_differential_equation" title="Nonlinear partial differential equation">nonlinear partial differential equation</a> for a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> dependent on two variables typically denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, involving the <a href="/wiki/Wave_operator" class="mw-redirect" title="Wave operator">wave operator</a> and the <a href="/wiki/Sine_and_cosine" title="Sine and cosine">sine</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>. </p><p>It was originally introduced by <a href="/wiki/Edmond_Bour" title="Edmond Bour">Edmond Bour</a> (<a href="#CITEREFBour1862">1862</a>) in the course of study of <a href="/wiki/Pseudosphere" title="Pseudosphere">surfaces of constant negative curvature</a> as the <a href="/wiki/Gauss%E2%80%93Codazzi_equation" class="mw-redirect" title="Gauss–Codazzi equation">Gauss–Codazzi equation</a> for surfaces of constant <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> −1 in <a href="/wiki/3-dimensional_space" class="mw-redirect" title="3-dimensional space">3-dimensional space</a>.<sup id="cite_ref-Bour1862_1-0" class="reference"><a href="#cite_note-Bour1862-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The equation was rediscovered by Frenkel and Kontorova (<a href="#CITEREFFrenkelKontorova1939">1939</a>) in their study of <a href="/wiki/Crystal_dislocation" class="mw-redirect" title="Crystal dislocation">crystal dislocations</a> known as the <a href="/wiki/Frenkel%E2%80%93Kontorova_model" title="Frenkel–Kontorova model">Frenkel–Kontorova model</a>.<sup id="cite_ref-FrenkelKontorova1939_2-0" class="reference"><a href="#cite_note-FrenkelKontorova1939-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>This equation attracted a lot of attention in the 1970s due to the presence of <a href="/wiki/Soliton" title="Soliton">soliton</a> solutions,<sup id="cite_ref-hirota_3-0" class="reference"><a href="#cite_note-hirota-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and is an example of an <a href="/wiki/Integrable_system" title="Integrable system">integrable PDE</a>. Among well-known integrable PDEs, the sine-Gordon equation is the only <i>relativistic</i> system due to its <a href="/wiki/Lorentz_invariance" class="mw-redirect" title="Lorentz invariance">Lorentz invariance</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Realizations_of_the_sine-Gordon_equation">Realizations of the sine-Gordon equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=1" title="Edit section: Realizations of the sine-Gordon equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Differential_geometry">Differential geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=2" title="Edit section: Differential geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is the first derivation of the equation, by Bour (1862). </p><p>There are two equivalent forms of the sine-Gordon equation. In the (<a href="/wiki/Real_number" title="Real number">real</a>) <i>space-time coordinates</i>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa3647f3b8798f94f0f2ac249637b0b709f3718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:2.843ex;" alt="{\displaystyle (x,t)}"></span>, the equation reads:<sup id="cite_ref-Rajaraman1989_4-0" class="reference"><a href="#cite_note-Rajaraman1989-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc58b170649144a87a5d6a9fb7e182a1c8789f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.924ex; height:2.676ex;" alt="{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}"></span></dd></dl> <p>where partial derivatives are denoted by subscripts. Passing to the <a href="/wiki/Light-cone_coordinates" title="Light-cone coordinates">light-cone coordinates</a> (<i>u</i>, <i>v</i>), akin to <i>asymptotic coordinates</i> where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaaee2273b7f411c8e48e59021ec7efbef003380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.349ex; height:5.176ex;" alt="{\displaystyle u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},}"></span></dd></dl> <p>the equation takes the form<sup id="cite_ref-Polyanin2004_5-0" class="reference"><a href="#cite_note-Polyanin2004-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{uv}=\sin \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{uv}=\sin \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff1774b42d6ab04d9d6958183cb457cbb6904b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.998ex; height:2.676ex;" alt="{\displaystyle \varphi _{uv}=\sin \varphi .}"></span></dd></dl> <p>This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of <a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">surfaces</a> of constant <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> <i>K</i> = −1, also called <a href="/wiki/Pseudospherical_surface" class="mw-redirect" title="Pseudospherical surface">pseudospherical surfaces</a>. </p><p>Consider an arbitrary pseudospherical surface. Across every point on the surface there are two <a href="/wiki/Asymptotic_curve" title="Asymptotic curve">asymptotic curves</a>. This allows us to construct a distinguished coordinate system for such a surface, in which <i>u</i> = constant, <i>v</i> = constant are the asymptotic lines, and the coordinates are incremented by the <a href="/wiki/Arc_length" title="Arc length">arc length</a> on the surface. At every point on the surface, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> be the angle between the asymptotic lines. </p><p>The <a href="/wiki/First_fundamental_form" title="First fundamental form">first fundamental form</a> of the surface is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> <mo>+</mo> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47719e4e6ddcecbad3a8deb87b50bf6d893b9c0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.015ex; height:3.176ex;" alt="{\displaystyle ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},}"></span></dd></dl> <p>and the <a href="/wiki/Second_fundamental_form" title="Second fundamental form">second fundamental form</a> is<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=N=0,M=\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>N</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=N=0,M=\sin \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8736c33e09da115941acf04b6f710179380ce3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.343ex; height:2.676ex;" alt="{\displaystyle L=N=0,M=\sin \varphi }"></span>and the <a href="/wiki/Gauss%E2%80%93Codazzi_equation" class="mw-redirect" title="Gauss–Codazzi equation">Gauss–Codazzi equation</a> is<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{uv}=\sin \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{uv}=\sin \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff1774b42d6ab04d9d6958183cb457cbb6904b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.998ex; height:2.676ex;" alt="{\displaystyle \varphi _{uv}=\sin \varphi .}"></span>Thus, any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily <a href="/wiki/Singular_curve" class="mw-redirect" title="Singular curve">singular</a> due to the <a href="/wiki/Hilbert%27s_theorem_(differential_geometry)" title="Hilbert's theorem (differential geometry)">Hilbert embedding theorem</a>. In the simplest case, <i>the</i> <a href="/wiki/Pseudosphere" title="Pseudosphere">pseudosphere</a>, also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator. </p><p>Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to <a href="/wiki/Rigid_transformation" title="Rigid transformation">rigid transformations</a>. There is a theorem, sometimes called the <i>fundamental theorem of surfaces</i>, that if a pair of matrix-valued bilinear forms satisfy the Gauss–Codazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space. Solutions to the sine-Gordon equation can be used to construct such matrices by using the forms obtained above. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Deforming_a_pseudosphere_to_Dini%27s_surface.gif" class="mw-file-description"><img alt="A pseudosphere is deformed to a Dini surface through the Lie transform" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Deforming_a_pseudosphere_to_Dini%27s_surface.gif/500px-Deforming_a_pseudosphere_to_Dini%27s_surface.gif" decoding="async" width="500" height="281" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fe/Deforming_a_pseudosphere_to_Dini%27s_surface.gif 1.5x" data-file-width="600" data-file-height="337" /></a><figcaption>Lie transform applied to pseudosphere to obtain a <a href="/wiki/Dini%27s_surface" title="Dini's surface">Dini surface</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="New_solutions_from_old">New solutions from old</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=3" title="Edit section: New solutions from old"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by <a href="/wiki/Luigi_Bianchi" title="Luigi Bianchi">Bianchi</a> and <a href="/wiki/Albert_Victor_B%C3%A4cklund" title="Albert Victor Bäcklund">Bäcklund</a> led to the discovery of <a href="/wiki/B%C3%A4cklund_transformation" class="mw-redirect" title="Bäcklund transformation">Bäcklund transformations</a>. Another transformation of pseudospherical surfaces is the <a href="/wiki/Squeeze_mapping#Lie_transform" title="Squeeze mapping">Lie transform</a> introduced by <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> in 1879, which corresponds to <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">Lorentz boosts</a> for solutions of the sine-Gordon equation.<sup id="cite_ref-terng_6-0" class="reference"><a href="#cite_note-terng-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since the sine-Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign-change comes down to a choice of direction for the normal to the surface. New solutions can be found by translating the solution: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is a solution, then so is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi +2n\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi +2n\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e4eb280aaa44c22c6db1dbad8f879908848974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.25ex; height:2.676ex;" alt="{\displaystyle \varphi +2n\pi }"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> an integer. </p> <div class="mw-heading mw-heading3"><h3 id="Frenkel–Kontorova_model"><span id="Frenkel.E2.80.93Kontorova_model"></span>Frenkel–Kontorova model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=4" title="Edit section: Frenkel–Kontorova model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Frenkel%E2%80%93Kontorova_model" title="Frenkel–Kontorova model">Frenkel–Kontorova model</a></div> <div class="mw-heading mw-heading3"><h3 id="A_mechanical_model">A mechanical model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=5" title="Edit section: A mechanical model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sine_gordon_5.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Sine_gordon_5.gif/220px-Sine_gordon_5.gif" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Sine_gordon_5.gif/330px-Sine_gordon_5.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d5/Sine_gordon_5.gif 2x" data-file-width="388" data-file-height="288" /></a><figcaption>A line of pendula, with a "breather pattern" oscillating in the middle. Unfortunately, the picture is drawn with gravity pointing <i>up</i>.</figcaption></figure> <p>Consider a line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>, then schematically, the dynamics of the line of pendulum follows Newton's second law:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {m\varphi _{tt}} _{\text{mass times acceleration}}=\underbrace {T\varphi _{xx}} _{\text{tension}}-\underbrace {mg\sin \varphi } _{\text{gravity}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>m</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mass times acceleration</mtext> </mrow> </munder> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>T</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tension</mtext> </mrow> </munder> <mo>−</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>m</mi> <mi>g</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>gravity</mtext> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {m\varphi _{tt}} _{\text{mass times acceleration}}=\underbrace {T\varphi _{xx}} _{\text{tension}}-\underbrace {mg\sin \varphi } _{\text{gravity}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a4498f6ba13026ce2672500829b29f28938f5a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; margin-right: -0.028ex; width:36.533ex; height:6.343ex;" alt="{\displaystyle \underbrace {m\varphi _{tt}} _{\text{mass times acceleration}}=\underbrace {T\varphi _{xx}} _{\text{tension}}-\underbrace {mg\sin \varphi } _{\text{gravity}}}"></span>and this is the sine-Gordon equation, after scaling time and distance appropriately. </p><p>Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\varphi _{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\varphi _{xx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccb280596f99f10f47e1b69e2f0e9c77631973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.269ex; height:2.676ex;" alt="{\displaystyle T\varphi _{xx}}"></span>, but more accurately <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe964e2a7f73c8f5a86f12fedeb638ed616d288f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.751ex; height:3.343ex;" alt="{\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2}}"></span>. However this does give an intuitive picture for the sine-gordon equation. One can produce exact mechanical realizations of the sine-gordon equation by more complex methods.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Naming">Naming</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=6" title="Edit section: Naming"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The name "sine-Gordon equation" is a pun on the well-known <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a> in physics:<sup id="cite_ref-Rajaraman1989_4-1" class="reference"><a href="#cite_note-Rajaraman1989-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>φ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e021e25764fc696568de1e4015aaa7eb5bc082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.681ex; height:2.676ex;" alt="{\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi =0.}"></span></dd></dl> <p>The sine-Gordon equation is the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a> of the field whose <a href="/wiki/Lagrangian_density" class="mw-redirect" title="Lagrangian density">Lagrangian density</a> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\text{SG}}(\varphi )={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>SG</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{SG}}(\varphi )={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a478426bbe4d0fa566c582675da893c5bfdfd85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.892ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}_{\text{SG}}(\varphi )={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .}"></span></dd></dl> <p>Using the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion of the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> in the Lagrangian, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8323a8ef51abcf384daee81fde7eb0c82908ba08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.174ex; height:7.009ex;" alt="{\displaystyle \cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}},}"></span></dd></dl> <p>it can be rewritten as the <a href="/wiki/Scalar_field_theory#Linear_.28free.29_theory" title="Scalar field theory">Klein–Gordon Lagrangian</a> plus higher-order terms: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{SG}}(\varphi )&={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2}}{2}}+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}\\&={\mathcal {L}}_{\text{KG}}(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>SG</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>KG</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{SG}}(\varphi )&={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2}}{2}}+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}\\&={\mathcal {L}}_{\text{KG}}(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfce1430ead5842bf343b5614773d971002c0cc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:44.214ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{SG}}(\varphi )&={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2}}{2}}+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}\\&={\mathcal {L}}_{\text{KG}}(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Soliton_solutions">Soliton solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=7" title="Edit section: Soliton solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An interesting feature of the sine-Gordon equation is the existence of <a href="/wiki/Soliton" title="Soliton">soliton</a> and multisoliton solutions. </p> <div class="mw-heading mw-heading3"><h3 id="1-soliton_solutions">1-soliton solutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=8" title="Edit section: 1-soliton solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sine-Gordon equation has the following 1-<a href="/wiki/Soliton" title="Soliton">soliton</a> solutions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>soliton</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4928ec052e8b0b3283d56b317347dc104f413e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.269ex; height:4.843ex;" alt="{\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b218de62f8789dbcdc521514b131082474d28c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.1ex; height:5.676ex;" alt="{\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}},}"></span></dd></dl> <p>and the slightly more general form of the equation is assumed: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c3758e771f53122b458fbac6693cbaa43f2bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.406ex; height:3.176ex;" alt="{\displaystyle \varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.}"></span></dd></dl> <p>The 1-soliton solution for which we have chosen the positive root for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is called a <i>kink</i> and represents a twist in the variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> which takes the system from one constant solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \varphi =0}"></span> to an adjacent constant solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb1052bf46b51a691a80a88b2524e4973314776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.113ex; height:2.676ex;" alt="{\displaystyle \varphi =2\pi }"></span>. The states <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \cong 2\pi n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>≅</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \cong 2\pi n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11bfad57e16bbe3c4537baa36126b64b64b2ea78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.508ex; height:2.676ex;" alt="{\displaystyle \varphi \cong 2\pi n}"></span> are known as vacuum states, as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is called an <i>antikink</i>. The form of the 1-soliton solutions can be obtained through application of a <a href="/wiki/B%C3%A4cklund_transform" title="Bäcklund transform">Bäcklund transform</a> to the trivial (vacuum) solution and the integration of the resulting first-order differentials: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ac72e2b3a596ef32d8ad57a5cd6e30470f12ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.497ex; height:5.509ex;" alt="{\displaystyle \varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2}},}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta }}\sin {\frac {\varphi '-\varphi }{2}}{\text{ with }}\varphi =\varphi _{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>β</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> with </mtext> </mrow> <mi>φ</mi> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta }}\sin {\frac {\varphi '-\varphi }{2}}{\text{ with }}\varphi =\varphi _{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739f57e4ef5ae9fd36d62c570fd167d2f5436170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.183ex; height:6.009ex;" alt="{\displaystyle \varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta }}\sin {\frac {\varphi '-\varphi }{2}}{\text{ with }}\varphi =\varphi _{0}=0}"></span></dd></dl> <p>for all time. </p><p>The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio Rubinstein in 1970.<sup id="cite_ref-Rubinstein1970_8-0" class="reference"><a href="#cite_note-Rubinstein1970-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Here we take a clockwise (<a href="/wiki/Right-hand_rule" title="Right-hand rule">left-handed</a>) twist of the elastic ribbon to be a kink with topological charge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{K}}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\text{K}}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a9879f3a10c20efe55756ee0c46f6fec4070e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.67ex; height:2.509ex;" alt="{\displaystyle \theta _{\text{K}}=-1}"></span>. The alternative counterclockwise (<a href="/wiki/Right-hand_rule" title="Right-hand rule">right-handed</a>) twist with topological charge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{AK}}=+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>AK</mtext> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\text{AK}}=+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43d13bcdad4ca837ba2f9e3e2a04ec6518221db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.903ex; height:2.509ex;" alt="{\displaystyle \theta _{\text{AK}}=+1}"></span> will be an antikink. </p> <table> <tbody><tr> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_1.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3f/Sine_gordon_1.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption>Traveling <i>kink</i> soliton represents a propagating clockwise twist.<sup id="cite_ref-Georgiev2004_9-0" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c2/Sine_gordon_2.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption>Traveling <i>antikink</i> soliton represents a propagating counterclockwise twist.<sup id="cite_ref-Georgiev2004_9-1" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td></tr></tbody></table> <figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:Static_one-soliton.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Static_one-soliton.png/474px-Static_one-soliton.png" decoding="async" width="474" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Static_one-soliton.png/711px-Static_one-soliton.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/de/Static_one-soliton.png 2x" data-file-width="720" data-file-height="456" /></a><figcaption>Static 1-soliton solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\arctan e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\arctan e^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb7f71f006aebf8129738b9e1f7d17f756fd630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.659ex; height:2.343ex;" alt="{\displaystyle 4\arctan e^{x}}"></span></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="2-soliton_solutions">2-soliton solutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=9" title="Edit section: 2-soliton solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Multi-<a href="/wiki/Soliton" title="Soliton">soliton</a> solutions can be obtained through continued application of the <a href="/wiki/B%C3%A4cklund_transform" title="Bäcklund transform">Bäcklund transform</a> to the 1-soliton solution, as prescribed by a <a href="/w/index.php?title=Bianchi_lattice&action=edit&redlink=1" class="new" title="Bianchi lattice (page does not exist)">Bianchi lattice</a> relating the transformed results.<sup id="cite_ref-Rogers2002_10-0" class="reference"><a href="#cite_note-Rogers2002-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase shift</a>. Since the colliding solitons recover their <a href="/wiki/Velocity" title="Velocity">velocity</a> and <a href="/wiki/Shape" title="Shape">shape</a>, such an interaction is called an <a href="/wiki/Elastic_collision" title="Elastic collision">elastic collision</a>. </p><p>The kink-kink solution is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095aec0aad57491110aa42bf9006a3403dfa6abe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:38.784ex; height:9.843ex;" alt="{\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}"></span> </p><p>while the kink-antikink solution is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5191e8ba6a3157c40b483ab700a203436fe9bcc6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:40.272ex; height:9.843ex;" alt="{\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}"></span> </p> <table> <tbody><tr> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d3/Sine_gordon_3.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption><i>Antikink-kink</i> collision.<sup id="cite_ref-Georgiev2004_9-2" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_4.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/dd/Sine_gordon_4.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption><i>Kink-kink</i> collision.<sup id="cite_ref-Georgiev2004_9-3" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td></tr></tbody></table> <p>Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a <i><a href="/wiki/Breather" title="Breather">breather</a></i>. There are known three types of breathers: <i>standing breather</i>, <i>traveling large-amplitude breather</i>, and <i>traveling small-amplitude breather</i>.<sup id="cite_ref-mir_11-0" class="reference"><a href="#cite_note-mir-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The standing breather solution is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thickmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ω</mi> <mspace width="thickmathspace" /> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb7300d39df60dcd4b94fc9a194a091b9e31c57" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.304ex; height:7.509ex;" alt="{\displaystyle \varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).}"></span> </p> <table> <tbody><tr> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_5.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d5/Sine_gordon_5.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption>The <i>standing breather</i> is an oscillating coupled kink-antikink soliton.<sup id="cite_ref-Georgiev2004_9-4" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_6.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/24/Sine_gordon_6.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption><i>Large-amplitude moving breather</i>.<sup id="cite_ref-Georgiev2004_9-5" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td></tr></tbody></table> <table> <tbody><tr> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_7.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/47/Sine_gordon_7.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption><i>Small-amplitude moving breather</i> –  looks exotic, but essentially has a breather envelope.<sup id="cite_ref-Georgiev2004_9-6" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="3-soliton_solutions">3-soliton solutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=10" title="Edit section: 3-soliton solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather, the shift of the breather <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b280feedc7f871327a45c27148156b135ff4b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.332ex; height:2.509ex;" alt="{\displaystyle \Delta _{\text{B}}}"></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\text{B}}={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K}}^{2})}}}{\sqrt {1-\omega ^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>artanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\text{B}}={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K}}^{2})}}}{\sqrt {1-\omega ^{2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90b59a9ac368a0ea543eec070df76106036ebcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.663ex; height:9.009ex;" alt="{\displaystyle \Delta _{\text{B}}={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K}}^{2})}}}{\sqrt {1-\omega ^{2}}}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{K}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfe0135a8b80e1c05127cb8cdec869ad45d6359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.009ex;" alt="{\displaystyle v_{\text{K}}}"></span> is the velocity of the kink, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the breather's frequency.<sup id="cite_ref-mir_11-1" class="reference"><a href="#cite_note-mir-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> If the old position of the standing breather is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, after the collision the new position will be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}+\Delta _{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}+\Delta _{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4215d15669ab7390355f31b1bff556dcb09aef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.556ex; height:2.509ex;" alt="{\displaystyle x_{0}+\Delta _{\text{B}}}"></span>. </p> <table> <tbody><tr> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_8.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/8a/Sine_gordon_8.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption>Collision of <i>moving kink</i> and <i>standing breather</i>.<sup id="cite_ref-Georgiev2004_9-7" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td> <td><figure typeof="mw:File/Frame"><a href="/wiki/File:Sine_gordon_9.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/aa/Sine_gordon_9.gif" decoding="async" width="388" height="288" class="mw-file-element" data-file-width="388" data-file-height="288" /></a><figcaption>Collision of <i>moving antikink</i> and <i>standing breather</i>.<sup id="cite_ref-Georgiev2004_9-8" class="reference"><a href="#cite_note-Georgiev2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Bäcklund_transformation"><span id="B.C3.A4cklund_transformation"></span>Bäcklund transformation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=11" title="Edit section: Bäcklund transformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/B%C3%A4cklund_transform" title="Bäcklund transform">Bäcklund transform</a></div> <p>Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is a solution of the sine-Gordon equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{uv}=\sin \varphi .\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{uv}=\sin \varphi .\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a68841d7773ffd6854524f147a93199cf1da7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.385ex; height:2.676ex;" alt="{\displaystyle \varphi _{uv}=\sin \varphi .\,}"></span></dd></dl> <p>Then the system </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2}}{\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a}}\sin {\Bigl (}{\frac {\psi -\varphi }{2}}{\Bigr )}\end{aligned}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ψ</mi> <mo>+</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>a</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ψ</mi> <mo>−</mo> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2}}{\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a}}\sin {\Bigl (}{\frac {\psi -\varphi }{2}}{\Bigr )}\end{aligned}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7782f68eb41b8921ed2612611014f3b2771438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; margin-right: -0.387ex; width:29.302ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2}}{\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a}}\sin {\Bigl (}{\frac {\psi -\varphi }{2}}{\Bigr )}\end{aligned}}\,\!}"></span></dd></dl> <p>where <i>a</i> is an arbitrary parameter, is solvable for a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform, as both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> are solutions to the same equation, that is, the sine-Gordon equation. </p><p>By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation. </p><p>For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is the trivial solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \equiv 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>≡</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \equiv 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06f4ce465ef11eabac7e87ba2482474301707f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \varphi \equiv 0}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> is the one-soliton solution with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> related to the boost applied to the soliton. </p> <div class="mw-heading mw-heading2"><h2 id="Topological_charge_and_energy">Topological charge and energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=12" title="Edit section: Topological charge and energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>topological charge</b> or <b>winding number</b> of a solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0119482d90c6139c5146d3c2429cdac0ddf28218" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.416ex; height:5.676ex;" alt="{\displaystyle N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].}"></span> The <b>energy</b> of a solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>d</mi> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5c2cf8acc77ae5d2fd3611d17cee39624710e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.007ex; height:6.176ex;" alt="{\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)}"></span>where a constant energy density has been added so that the potential is non-negative. With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as interactions. </p><p>The topological charge is conserved if the energy is finite. The topological charge does not determine the solution, even up to Lorentz boosts. Both the trivial solution and the soliton-antisoliton pair solution have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac661e6a8863869cdec476a5f7923be96ee0e97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=0}"></span>. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Zero-curvature_formulation">Zero-curvature formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=13" title="Edit section: Zero-curvature formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sine-Gordon equation is equivalent to the <a href="/wiki/Curvature_form" title="Curvature form">curvature</a> of a particular <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"></span>-<a href="/wiki/Principal_connection" class="mw-redirect" title="Principal connection">connection</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> being equal to zero.<sup id="cite_ref-SIT_12-0" class="reference"><a href="#cite_note-SIT-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Explicitly, with coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eadf12294edccd7a29c99cfc1765e4a14bf47e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.301ex; height:2.843ex;" alt="{\displaystyle (u,v)}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>, the connection components <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9277f5286335ab99c040c9c9151ab752d3bedc49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.967ex; height:2.843ex;" alt="{\displaystyle A_{\mu }}"></span> are given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>i</mi> <mi>λ</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mi>λ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>λ</mi> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92cf3ba0d3901069a5fc35b59709bbdc6986e0d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.471ex; height:7.509ex;" alt="{\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf179bbd8861eacf7de7e430d75d87050d5f558f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:64.056ex; height:7.843ex;" alt="{\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},}"></span> where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}"></span> are the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. Then the zero-curvature equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1df78e5436db77fbcade01f1384a9b8215a448b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.317ex; height:2.843ex;" alt="{\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0}"></span> </p><p>is equivalent to the sine-Gordon equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{uv}=\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{uv}=\sin \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c59d0752d87183dc929aa2221b61f159f9d83c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.351ex; height:2.676ex;" alt="{\displaystyle \varphi _{uv}=\sin \varphi }"></span>. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2f7a703f38b35504f21319fbc4ee6a75dfd35d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.305ex; height:3.009ex;" alt="{\displaystyle F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]}"></span>. </p><p>The pair of matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{u}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{u}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9fa2f64881642cb91a4cb47410d4a2abc46a639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.916ex; height:2.509ex;" alt="{\displaystyle A_{u}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd85529417776f1ae43598d068b5ed468a4b704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.773ex; height:2.509ex;" alt="{\displaystyle A_{v}}"></span> are also known as a <a href="/wiki/Lax_pair" title="Lax pair">Lax pair</a> for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation. </p> <div class="mw-heading mw-heading2"><h2 id="Related_equations">Related equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=14" title="Edit section: Related equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="sinh-Gordon_equation"></span><span class="vanchor-text">sinh-Gordon equation</span></span></b> is given by<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{xx}-\varphi _{tt}=\sinh \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{xx}-\varphi _{tt}=\sinh \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953437b55f6e28b04610c7042b346679e8a595fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.214ex; height:2.676ex;" alt="{\displaystyle \varphi _{xx}-\varphi _{tt}=\sinh \varphi .}"></span></dd></dl> <p>This is the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a> of the <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea05c2a0a19e1cb19de817a56be8f0c1f37e416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.416ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .}"></span></dd></dl> <p>Another closely related equation is the <b>elliptic sine-Gordon equation</b> or <b>Euclidean sine-Gordon equation</b>, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{xx}+\varphi _{yy}=\sin \varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{xx}+\varphi _{yy}=\sin \varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e43260807fdf3754d75dbe28612537325a53ff4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.368ex; height:2.843ex;" alt="{\displaystyle \varphi _{xx}+\varphi _{yy}=\sin \varphi ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is now a function of the variables <i>x</i> and <i>y</i>. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine-Gordon equation by the <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> (or <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a>) <i>y</i> = i<i>t</i>. </p><p>The <b>elliptic sinh-Gordon equation</b> may be defined in a similar way. </p><p>Another similar equation comes from the Euler–Lagrange equation for <a href="/wiki/Liouville_field_theory" title="Liouville field theory">Liouville field theory</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>φ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9a62a2b228cea25d0dcad66b7194839bca2577" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.533ex; height:3.176ex;" alt="{\displaystyle \varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.}"></span> </p><p>A generalization is given by <a href="/wiki/Toda_field_theory" title="Toda field theory">Toda field theory</a>.<sup id="cite_ref-Yuanxi_14-0" class="reference"><a href="#cite_note-Yuanxi-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> More precisely, Liouville field theory is the Toda field theory for the finite <a href="/wiki/Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebra</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d39fbc95eae134384e3b799cc509e1708b643cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {sl}}_{2}}"></span>, while sin(h)-Gordon is the Toda field theory for the <a href="/wiki/Affine_Kac%E2%80%93Moody_algebra" class="mw-redirect" title="Affine Kac–Moody algebra">affine Kac–Moody algebra</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathfrak {sl}}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathfrak {sl}}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1713b74df7c26254685797c67d1ad3f9a32de2eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:2.776ex; height:3.176ex;" alt="{\displaystyle {\hat {\mathfrak {sl}}}_{2}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Infinite_volume_and_on_a_half_line">Infinite volume and on a half line</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=15" title="Edit section: Infinite volume and on a half line"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can also consider the sine-Gordon model on a circle,<sup id="cite_ref-McKean1981_15-0" class="reference"><a href="#cite_note-McKean1981-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> on a line segment, or on a half line.<sup id="cite_ref-Bowcock2007_16-0" class="reference"><a href="#cite_note-Bowcock2007-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> It is possible to find boundary conditions which preserve the integrability of the model.<sup id="cite_ref-Bowcock2007_16-1" class="reference"><a href="#cite_note-Bowcock2007-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> On a half line the spectrum contains <i>boundary bound states</i> in addition to the solitons and breathers.<sup id="cite_ref-Bowcock2007_16-2" class="reference"><a href="#cite_note-Bowcock2007-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_sine-Gordon_model">Quantum sine-Gordon model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=16" title="Edit section: Quantum sine-Gordon model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> the sine-Gordon model contains a parameter that can be identified with the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of <a href="/wiki/Breather" title="Breather">breathers</a>.<sup id="cite_ref-Korepin1979_17-0" class="reference"><a href="#cite_note-Korepin1979-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Takada1981_18-0" class="reference"><a href="#cite_note-Takada1981-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bogoliubov1985_19-0" class="reference"><a href="#cite_note-Bogoliubov1985-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The number of the breathers depends on the value of the parameter. Multiparticle production cancels on mass shell. </p><p>Semi-classical quantization of the sine-Gordon model was done by <a href="/wiki/Ludwig_Faddeev" class="mw-redirect" title="Ludwig Faddeev">Ludwig Faddeev</a> and <a href="/wiki/Vladimir_Korepin" title="Vladimir Korepin">Vladimir Korepin</a>.<sup id="cite_ref-Faddeev1978_20-0" class="reference"><a href="#cite_note-Faddeev1978-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The exact quantum <a href="/wiki/Scattering_matrix" class="mw-redirect" title="Scattering matrix">scattering matrix</a> was discovered by <a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Alexander Zamolodchikov</a>.<sup id="cite_ref-Zamolodchikov1978_21-0" class="reference"><a href="#cite_note-Zamolodchikov1978-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> This model is <a href="/wiki/S-duality" title="S-duality">S-dual</a> to the <a href="/wiki/Thirring_model" title="Thirring model">Thirring model</a>, as discovered by <a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman</a>. <sup id="cite_ref-coleman_22-0" class="reference"><a href="#cite_note-coleman-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> This is sometimes known as the Coleman correspondence and serves as an example of boson-fermion correspondence in the interacting case. This article also showed that the constants appearing in the model behave nicely under <a href="/wiki/Renormalization" title="Renormalization">renormalization</a>: there are three parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{0},\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{0},\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87413edba87917e376c099d3686b30d5f3b80084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.908ex; height:2.509ex;" alt="{\displaystyle \alpha _{0},\beta }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407f2fbc64e21bb432f397fdb816de0f5083d900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{0}}"></span>. Coleman showed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a214eff2fcc322f780dd8837e7472b0edb994a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{0}}"></span> receives only a multiplicative correction, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407f2fbc64e21bb432f397fdb816de0f5083d900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{0}}"></span> receives only an additive correction, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> is not renormalized. Further, for a critical, non-zero value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={\sqrt {4\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <mi>π</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={\sqrt {4\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b14bace1099a327e1422cdef4a4d15c6dc3e8a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:3.009ex;" alt="{\displaystyle \beta ={\sqrt {4\pi }}}"></span>, the theory is in fact dual to a <i>free</i> massive <a href="/wiki/Dirac_equation#Lagrangian_formulation" title="Dirac equation">Dirac field theory</a>. </p><p>The quantum sine-Gordon equation should be modified so the exponentials become <a href="/wiki/Vertex_operator" class="mw-redirect" title="Vertex operator">vertex operators</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{QsG}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2}}m_{0}^{2}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mi>s</mi> <mi>G</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>φ</mi> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{QsG}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2}}m_{0}^{2}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3110faf23e7261c450784e3a47eb84cdc0d78d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.183ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}_{QsG}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2}}m_{0}^{2}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\beta }=:e^{i\beta \varphi }:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> <mi>φ</mi> </mrow> </msup> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\beta }=:e^{i\beta \varphi }:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9b34a0119d769c9247097f1378ffc11b3cb580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.467ex; height:3.343ex;" alt="{\displaystyle V_{\beta }=:e^{i\beta \varphi }:}"></span>, where the semi-colons denote <a href="/wiki/Normal_ordering" class="mw-redirect" title="Normal ordering">normal ordering</a>. A possible mass term is included. </p> <div class="mw-heading mw-heading3"><h3 id="Regimes_of_renormalizability">Regimes of renormalizability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=17" title="Edit section: Regimes of renormalizability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For different values of the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a79f17c76984dead8b0c072b4fdf537b13895e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.391ex; height:3.009ex;" alt="{\displaystyle \beta ^{2}}"></span>, the <a href="/wiki/Renormalization" title="Renormalization">renormalizability</a> properties of the sine-Gordon theory change.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The identification of these regimes is attributed to <a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Jürg Fröhlich</a>. </p><p>The <b>finite regime</b> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{2}<4\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>4</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{2}<4\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c657eeceaa04ae7bce987b3c2f7916d13e3a0fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.984ex; height:3.009ex;" alt="{\displaystyle \beta ^{2}<4\pi }"></span>, where no <a href="/wiki/Counterterm" class="mw-redirect" title="Counterterm">counterterms</a> are needed to render the theory well-posed. The <b>super-renormalizable regime</b> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi <\beta ^{2}<8\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <mo><</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>8</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi <\beta ^{2}<8\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46296712400fcae10f6e5fac3cfb002602868601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.577ex; height:3.009ex;" alt="{\displaystyle 4\pi <\beta ^{2}<8\pi }"></span>, where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{n+1}}8\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mn>8</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{n+1}}8\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c36bb8c20ca11b761ab2987da1b6166648323593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.728ex; height:4.843ex;" alt="{\displaystyle {\frac {n}{n+1}}8\pi }"></span> passed.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{2}>8\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>8</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{2}>8\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c9dbe2207b5d1f4afc9f067b4189c09c3a7592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.984ex; height:3.009ex;" alt="{\displaystyle \beta ^{2}>8\pi }"></span>, the theory becomes ill-defined (Coleman <a href="#CITEREFColeman1975">1975</a>). The boundary values are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{2}=4\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{2}=4\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac0cdbaa36ab9358f89c1fb7a05f591fe2917e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.984ex; height:3.009ex;" alt="{\displaystyle \beta ^{2}=4\pi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{2}=8\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>8</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{2}=8\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96a3bb8d8e052835208e95af0d659535fd6b644d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.984ex; height:3.009ex;" alt="{\displaystyle \beta ^{2}=8\pi }"></span>, which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self-dual point, where the vertex operators form an <a href="/wiki/Affine_Kac%E2%80%93Moody_algebra" class="mw-redirect" title="Affine Kac–Moody algebra">affine sl<sub>2</sub> subalgebra</a>, and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable). </p> <div class="mw-heading mw-heading2"><h2 id="Stochastic_sine-Gordon_model">Stochastic sine-Gordon model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=18" title="Edit section: Stochastic sine-Gordon model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>stochastic</b> or <b>dynamical sine-Gordon model</b> has been studied by <a href="/wiki/Martin_Hairer" title="Martin Hairer">Martin Hairer</a> and Hao Shen <sup id="cite_ref-hairer-shen_25-0" class="reference"><a href="#cite_note-hairer-shen-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting. </p><p>The equation is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}u={\frac {1}{2}}\Delta u+c\sin(\beta u+\theta )+\xi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>c</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>u</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}u={\frac {1}{2}}\Delta u+c\sin(\beta u+\theta )+\xi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe26ebe17fff5a2036c83938a7fe16603498235" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.762ex; height:5.176ex;" alt="{\displaystyle \partial _{t}u={\frac {1}{2}}\Delta u+c\sin(\beta u+\theta )+\xi ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c,\beta ,\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,\beta ,\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2bea911a086ba700b9c6ebd39be59e05edf17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.497ex; height:2.509ex;" alt="{\displaystyle c,\beta ,\theta }"></span> are real-valued constants, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"></span> is space-time <a href="/wiki/White_noise" title="White noise">white noise</a>. The space dimension is fixed to 2. In the proof of existence of solutions, the thresholds <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{2}={\frac {n}{n+1}}8\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mn>8</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{2}={\frac {n}{n+1}}8\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d2bcc4c94761e3765108d4f529de45801e49c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.218ex; height:4.843ex;" alt="{\displaystyle \beta ^{2}={\frac {n}{n+1}}8\pi }"></span> again play a role in determining convergence of certain terms. </p> <div class="mw-heading mw-heading2"><h2 id="Supersymmetric_sine-Gordon_model">Supersymmetric sine-Gordon model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=19" title="Edit section: Supersymmetric sine-Gordon model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A supersymmetric extension of the sine-Gordon model also exists.<sup id="cite_ref-Inami1995_26-0" class="reference"><a href="#cite_note-Inami1995-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Integrability preserving boundary conditions for this extension can be found as well.<sup id="cite_ref-Inami1995_26-1" class="reference"><a href="#cite_note-Inami1995-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Physical_applications">Physical applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=20" title="Edit section: Physical applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sine-Gordon model arises as the continuum limit of the <a href="/wiki/Frenkel%E2%80%93Kontorova_model" title="Frenkel–Kontorova model">Frenkel–Kontorova model</a> which models crystal dislocations. </p><p>Dynamics in <a href="/wiki/Long_Josephson_junction" title="Long Josephson junction">long Josephson junctions</a> are well-described by the sine-Gordon equations, and conversely provide a useful experimental system for studying the sine-Gordon model.<sup id="cite_ref-MU_27-0" class="reference"><a href="#cite_note-MU-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>The sine-Gordon model is in the same <a href="/wiki/Universality_class" title="Universality class">universality class</a> as the <a href="/wiki/Effective_action" title="Effective action">effective action</a> for a <a href="/wiki/Coulomb_gas" title="Coulomb gas">Coulomb gas</a> of <a href="/wiki/Quantum_vortex" title="Quantum vortex">vortices</a> and anti-vortices in the continuous <a href="/wiki/Classical_XY_model" title="Classical XY model">classical XY model</a>, which is a model of magnetism.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Kosterlitz%E2%80%93Thouless_transition" class="mw-redirect" title="Kosterlitz–Thouless transition">Kosterlitz–Thouless transition</a> for vortices can therefore be derived from a <a href="/wiki/Renormalization_group" title="Renormalization group">renormalization group</a> analysis of the sine-Gordon field theory.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>The sine-Gordon equation also arises as the formal continuum limit of a different model of magnetism, the <a href="/wiki/Quantum_Heisenberg_model" title="Quantum Heisenberg model">quantum Heisenberg model</a>, in particular the XXZ model.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a></li> <li><a href="/wiki/Fluxon" title="Fluxon">Fluxon</a></li> <li><a href="/wiki/Shape_waves" title="Shape waves">Shape waves</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine-Gordon_equation&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-Bour1862-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bour1862_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBour1862" class="citation journal cs1">Bour, Edmond (1862). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k433694t">"Theorie de la deformation des surfaces"</a>. <i>Journal de l'École impériale polytechnique</i>. <b>22</b> (39): 1–148. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/55567842">55567842</a>.</cite><span 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.hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_field_theories" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output 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abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_field_theories" title="Template:Quantum field theories"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_field_theories" title="Template talk:Quantum field theories"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theories" title="Special:EditPage/Template:Quantum field theories"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_field_theories" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theories</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic QFT</a></li> <li><a href="/wiki/Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">Axiomatic QFT</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li> <li><a href="/wiki/Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">Two-dimensional conformal field theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Regular</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born%E2%80%93Infeld_model" title="Born–Infeld model">Born–Infeld</a></li> <li><a href="/wiki/Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li> <li><a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li> <li><a href="/wiki/Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li> <li><a href="/wiki/Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li> <li><a href="/wiki/Soler_model" title="Soler model">Soler</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li> <li><a href="/wiki/Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li> <li><a href="/wiki/Yukawa_interaction" title="Yukawa interaction">Yukawa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Low dimensional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li> <li><a href="/wiki/Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li> <li><a href="/wiki/Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li> <li><a href="/wiki/Schwinger_model" title="Schwinger model">Schwinger</a></li> <li><a class="mw-selflink selflink">Sine-Gordon</a></li> <li><a href="/wiki/Thirring_model" title="Thirring model">Thirring</a></li> <li><a href="/wiki/Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li> <li><a href="/wiki/Toda_field_theory" title="Toda field theory">Toda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Conformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li> <li><a href="/wiki/Liouville_field_theory" title="Liouville field theory">Liouville</a></li> <li><a href="/wiki/Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supersymmetric</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/4D_N_%3D_1_global_supersymmetry" title="4D N = 1 global supersymmetry">4D N = 1</a></li> <li><a href="/wiki/N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li> <li><a href="/wiki/Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li> <li><a href="/wiki/Super_QCD" title="Super QCD">Super QCD</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Superconformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/6D_(2,0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0)</a></li> <li><a href="/wiki/ABJM_superconformal_field_theory" title="ABJM superconformal field theory">ABJM</a></li> <li><a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supergravity</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pure_4D_N_%3D_1_supergravity" title="Pure 4D N = 1 supergravity">Pure 4D N = 1</a></li> <li><a href="/wiki/4D_N_%3D_1_supergravity" title="4D N = 1 supergravity">4D N = 1</a></li> <li><a href="/wiki/N_%3D_8_supergravity" title="N = 8 supergravity">4D N = 8</a></li> <li><a href="/wiki/Higher-dimensional_supergravity" title="Higher-dimensional supergravity">Higher dimensional</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">11D</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Topological</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BF_model" title="BF model">BF</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Particle theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chiral_model" title="Chiral model">Chiral</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi</a></li> <li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></li> <li><a href="/wiki/Stueckelberg_action" title="Stueckelberg action">Stueckelberg</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic string</a></li> <li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li> <li><a href="/wiki/On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_foam" title="Quantum foam">Quantum foam</a></li> <li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a> <ul><li><a href="/wiki/Template:Quantum_electrodynamics" title="Template:Quantum electrodynamics">links</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a> <ul><li><a href="/wiki/Template:Quantum_gravity" title="Template:Quantum gravity">links</a></li></ul></li> <li><a href="/wiki/Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li> <li><a href="/wiki/Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a> <ul><li><a href="/wiki/Template:Quantum_information" title="Template:Quantum information">links</a></li></ul></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>See also:</i> <span class="noviewer" typeof="mw:File"><span title="Template"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics">Template:Quantum mechanics topics</a></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Integrable_systems" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Integrable_systems" title="Template:Integrable systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Integrable_systems" title="Template talk:Integrable systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Integrable_systems" title="Special:EditPage/Template:Integrable systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Integrable_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Integrable_system" title="Integrable system">Integrable systems</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Geometric integrability</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frobenius_integrability" class="mw-redirect" title="Frobenius integrability">Frobenius integrability</a> <ul><li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li></ul></li> <li><a href="/wiki/Liouville%E2%80%93Arnold_theorem" title="Liouville–Arnold theorem"> Liouville integrability</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">In classical mechanics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li> <li><a href="/wiki/Central_force" title="Central force">Central force systems</a> <ul><li><a href="/wiki/Kepler_problem" title="Kepler problem">Kepler system</a></li> <li><a href="/wiki/Two_body_problem" class="mw-redirect" title="Two body problem">Two body problem</a></li></ul></li> <li><a href="/wiki/Lagrange,_Euler_and_Kovalevskaya_tops" class="mw-redirect" title="Lagrange, Euler and Kovalevskaya tops">Integrable tops</a> <ul><li><a href="/wiki/Euler_top" class="mw-redirect" title="Euler top">Euler</a></li> <li><a href="/wiki/Kovalevskaya_top" class="mw-redirect" title="Kovalevskaya top">Kovalevskaya</a></li> <li><a href="/wiki/Lagrange_top" class="mw-redirect" title="Lagrange top">Lagrange</a></li></ul></li> <li><a href="/wiki/Garnier_integrable_system" title="Garnier integrable system">Garnier integrable system</a></li> <li><a href="/wiki/Hitchin_system" title="Hitchin system">Hitchin system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Liouville%E2%80%93Arnold_theorem" title="Liouville–Arnold theorem">Liouville–Arnold theorem</a></li> <li><a href="/wiki/Action-angle_variables" class="mw-redirect" title="Action-angle variables">Action-angle variables</a></li> <li><a href="/wiki/Superintegrable_Hamiltonian_system" title="Superintegrable Hamiltonian system">Superintegrable Hamiltonian system</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">In quantum mechanics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">Quantum harmonic oscillator</a></li> <li><a href="/wiki/Hydrogen_atom" title="Hydrogen atom">Hydrogen atom</a></li> <li><a href="/wiki/P%C3%B6schl%E2%80%93Teller_potential" title="Pöschl–Teller potential">Pöschl–Teller potential</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Integrable PDEs/Classical integrable field theories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/KdV_equation" class="mw-redirect" title="KdV equation">KdV equation</a> <ul><li><a href="/wiki/KdV_hierarchy" title="KdV hierarchy">KdV hierarchy</a></li></ul></li> <li><a class="mw-selflink selflink">Sine-Gordon equation</a></li> <li><a href="/wiki/Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">Nonlinear Schrödinger equation</a></li> <li><a href="/wiki/Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu model</a></li> <li><a href="/wiki/Thirring_model" title="Thirring model">Thirring model</a></li> <li><a href="/wiki/Kadomtsev%E2%80%93Petviashvili_equation" title="Kadomtsev–Petviashvili equation">Kadomtsev–Petviashvili equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%A4cklund_transformation" class="mw-redirect" title="Bäcklund transformation">Bäcklund transformation</a></li> <li><a href="/wiki/Lax_pairs" class="mw-redirect" title="Lax pairs">Lax pairs</a></li> <li>Infinitely many <a href="/wiki/Integral_of_motion" class="mw-redirect" title="Integral of motion">integrals of motion</a></li> <li><a href="/wiki/Soliton" title="Soliton">Soliton solutions</a> <ul><li><a href="/wiki/Topological_soliton" class="mw-redirect" title="Topological soliton">Topological soliton</a></li></ul></li> <li><a href="/wiki/Inverse_scattering_transform" title="Inverse scattering transform">Inverse scattering transform</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">ASDYM as a master theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Yang%E2%80%93Mills_equations#Anti-self-duality_equations" title="Yang–Mills equations">Anti-self-dual Yang–Mills equations</a></li> <li><a href="/wiki/Twistor_correspondence" title="Twistor correspondence">Twistor correspondence</a></li> <li><a href="/wiki/Ward_conjecture" class="mw-redirect" title="Ward conjecture">Ward conjecture</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Integrable Quantum Field theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink-fragment" href="#Quantum_version"> Quantum Sine-Gordon</a></li> <li><a href="/w/index.php?title=Quantum_KdV&action=edit&redlink=1" class="new" title="Quantum KdV (page does not exist)">Quantum KdV</a></li> <li><a href="/wiki/Liouville_field_theory" title="Liouville field theory">Quantum Liouville</a></li> <li><a href="/wiki/Thirring_model" title="Thirring model">Thirring model</a></li> <li><a href="/wiki/Toda_field_theory" title="Toda field theory">Toda field theory</a></li> <li><a href="/wiki/Principal_chiral_model" class="mw-redirect" title="Principal chiral model">Principal chiral model</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Master_theories" scope="row" class="navbox-group" style="width:1%">Master theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Four-dimensional_Chern%E2%80%93Simons_theory" title="Four-dimensional Chern–Simons theory">Four-dimensional Chern–Simons theory</a> (Lagrangian)</li> <li><a href="/wiki/Garnier_integrable_system" title="Garnier integrable system">Affine Gaudin models</a> (Hamiltonian)</li> <li><a href="/wiki/Six-dimensional_holomorphic_Chern%E2%80%93Simons_theory" title="Six-dimensional holomorphic Chern–Simons theory">Six-dimensional holomorphic Chern–Simons theory</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Exactly solvable statistical <a href="/wiki/Lattice_models" class="mw-redirect" title="Lattice models">lattice models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> in one- and two-dimensions</li> <li><a href="/wiki/Ice-type_model" title="Ice-type model">Square ice model</a></li> <li><a href="/wiki/Eight-vertex_model" title="Eight-vertex model">Eight-vertex model</a></li> <li><a href="/wiki/Hard_hexagon_model" title="Hard hexagon model">Hard hexagon model</a></li> <li><a href="/wiki/Chiral_Potts_model" title="Chiral Potts model">Chiral Potts model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Exactly solvable quantum spin chains</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Heisenberg_model" title="Quantum Heisenberg model">Quantum Heisenberg model</a></li> <li><a href="/wiki/Gaudin_model" title="Gaudin model">Gaudin model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Algebraic <a href="/wiki/Bethe_ansatz" title="Bethe ansatz">Bethe ansatz</a></li> <li><a href="/wiki/Quantum_inverse_scattering_method" title="Quantum inverse scattering method">Quantum inverse scattering method</a></li> <li><a href="/wiki/Yang%E2%80%93Baxter_equation" title="Yang–Baxter equation">Yang–Baxter equation</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Contributors</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Classical mechanics and geometry</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Vladimir Arnold</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Ferdinand_Georg_Frobenius" title="Ferdinand Georg Frobenius">Ferdinand Georg Frobenius</a></li> <li><a href="/wiki/Nigel_Hitchin" title="Nigel Hitchin">Nigel Hitchin</a></li> <li><a href="/wiki/Sofia_Kovalevskaya" class="mw-redirect" title="Sofia Kovalevskaya">Sofia Kovalevskaya</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Joseph Liouville</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">PDEs</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clifford_S._Gardner" title="Clifford S. Gardner">Clifford S. Gardner</a></li> <li><a href="/wiki/John_M._Greene" title="John M. Greene">John M. Greene</a></li> <li><a href="/wiki/Martin_David_Kruskal" title="Martin David Kruskal">Martin David Kruskal</a></li> <li><a href="/wiki/Peter_Lax" title="Peter Lax">Peter Lax</a></li> <li><a href="/wiki/Robert_Miura" class="mw-redirect" title="Robert Miura">Robert Miura</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">IQFTs</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Alexander Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Alexei Zamolodchikov</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classical and quantum statistical lattices</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rodney_Baxter" title="Rodney Baxter">Rodney Baxter</a></li> <li><a href="/wiki/Ludvig_Faddeev" title="Ludvig Faddeev">Ludvig Faddeev</a></li> <li><a href="/wiki/Elliott_H._Lieb" title="Elliott H. Lieb">Elliott H. Lieb</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang Chen-Ning</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Sine-Gordon_equation&oldid=1228993777">https://en.wikipedia.org/w/index.php?title=Sine-Gordon_equation&oldid=1228993777</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Solitons" title="Category:Solitons">Solitons</a></li><li><a href="/wiki/Category:Differential_geometry" title="Category:Differential geometry">Differential geometry</a></li><li><a href="/wiki/Category:Surfaces" title="Category:Surfaces">Surfaces</a></li><li><a href="/wiki/Category:Exactly_solvable_models" title="Category:Exactly solvable models">Exactly solvable models</a></li><li><a href="/wiki/Category:Equations_of_physics" title="Category:Equations of physics">Equations of physics</a></li><li><a href="/wiki/Category:Mathematical_physics" title="Category:Mathematical physics">Mathematical physics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_containing_video_clips" title="Category:Articles containing video clips">Articles containing video clips</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 June 2024, at 07:49<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Sine-Gordon equation - Wikipedia