Ecuación de Hunter-Saxton - Wikipedia, la enciclopedia libre

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class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><p>En <a href="/wiki/F%C3%ADsica_matem%C3%A1tica" title="Física matemática">física matemática</a> , la <b>ecuación de Hunter-Saxton</b> <sup id="cite_ref-HS91_1-0" class="reference separada"><a href="#cite_note-HS91-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</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 (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8b12e6efac14d9b5b88fda64c9c54904d5ac64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.796ex; height:5.176ex;" alt="{\displaystyle (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}}"></span></dd></dl> <p>es una <a href="/wiki/Ecuaci%C3%B3n_en_derivadas_parciales" title="Ecuación en derivadas parciales">ecuación en derivadas parciales</a> <a href="/wiki/Sistema_hamiltoniano_integrable" title="Sistema hamiltoniano integrable"> integrable</a> que surge en el estudio teórico de <a href="/wiki/Cristal_l%C3%ADquido" title="Cristal líquido"> cristales líquidos</a> nemáticos. Si las moléculas en el cristal líquido están inicialmente alineadas y algunas de ellas se mueven ligeramente, esta alteración en la orientación se propagará a través del cristal. La ecuación de Hunter-Saxton describe ciertos aspectos de tales ondas de orientación. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Antecedentes_físicos"><span id="Antecedentes_f.C3.ADsicos"></span>Antecedentes físicos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecuaci%C3%B3n_de_Hunter-Saxton&action=edit&section=1" title="Editar sección: Antecedentes físicos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En los modelos de cristales líquidos considerados aquí, se supone que no hay flujo de fluido, por lo que sólo la <i>orientación</i> de las moléculas es de interés. Dentro de la <i>teoría del elástico continuo</i>, la orientación se describe mediante un campo de vectores unitarios <b>n</b>(<i>x</i>,<i>y</i>,<i>z</i>,<i>t</i>). Para los cristales líquidos nemáticos, no hay diferencia entre orientar una molécula en <b>n</b> direcciones o en −<b>n</b> direcciones y el campo vectorial <b>n</b> es entonces llamado <i>campo director</i>. </p><p>La densidad de energía potencial de un campo director se supone que viene dada por la energía funcional de <a href="/wiki/Carl_Wilhelm_Oseen" title="Carl Wilhelm Oseen">Oseen</a>–<a href="/w/index.php?title=Charles_Frank_(f%C3%ADsico)&action=edit&redlink=1" class="new" title="Charles Frank (físico) (aún no redactado)">Frank</a><sup id="cite_ref-2" class="reference separada"><a href="#cite_note-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</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 W(\mathbf {n} ,\nabla \mathbf {n} )={\frac {1}{2}}\left(\alpha (\nabla \cdot \mathbf {n} )^{2}+\beta (\mathbf {n} \cdot (\nabla \times \mathbf {n} ))^{2}+\gamma |\mathbf {n} \times (\nabla \times \mathbf {n} )|^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>β</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(\mathbf {n} ,\nabla \mathbf {n} )={\frac {1}{2}}\left(\alpha (\nabla \cdot \mathbf {n} )^{2}+\beta (\mathbf {n} \cdot (\nabla \times \mathbf {n} ))^{2}+\gamma |\mathbf {n} \times (\nabla \times \mathbf {n} )|^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/879f3f55e369fc74c6ad543e8cf7d50c41bb020b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.048ex; height:5.176ex;" alt="{\displaystyle W(\mathbf {n} ,\nabla \mathbf {n} )={\frac {1}{2}}\left(\alpha (\nabla \cdot \mathbf {n} )^{2}+\beta (\mathbf {n} \cdot (\nabla \times \mathbf {n} ))^{2}+\gamma |\mathbf {n} \times (\nabla \times \mathbf {n} )|^{2}\right),}"></span></dd></dl> <p>donde los coeficientes positivos <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, <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>, <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> son conocidos como los coeficientes elásticos de separación, torsión y flexión, respectivamente. La energía cinética se descuida a menudo debido a la alta <a href="/wiki/Viscosidad" title="Viscosidad">viscosidad</a> de los cristales líquidos. </p> <div class="mw-heading mw-heading2"><h2 id="Derivación_de_la_ecuación_de_Hunter-Saxton"><span id="Derivaci.C3.B3n_de_la_ecuaci.C3.B3n_de_Hunter-Saxton"></span>Derivación de la ecuación de Hunter-Saxton</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecuaci%C3%B3n_de_Hunter-Saxton&action=edit&section=2" title="Editar sección: Derivación de la ecuación de Hunter-Saxton"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hunter y Saxton<sup id="cite_ref-HS91_1-1" class="reference separada"><a href="#cite_note-HS91-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> investigaron el caso en el que se ignora la amortiguación viscosa y se incluye un término de <a href="/wiki/Energ%C3%ADa_cin%C3%A9tica" title="Energía cinética">energía cinética</a> en el modelo. Entonces las ecuaciones rectoras para la dinámica del campo director son las <a href="/wiki/Ecuaciones_de_Euler-Lagrange" title="Ecuaciones de Euler-Lagrange">ecuaciones de Euler-Lagrange</a> para el <a href="/wiki/Lagrangiano" title="Lagrangiano">lagrangiano</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}}\left|{\frac {\partial \mathbf {n} }{\partial t}}\right|^{2}-W(\mathbf {n} ,\nabla \mathbf {n} )-{\frac {\lambda }{2}}(1-|\mathbf {n} |^{2}),}"> <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> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left|{\frac {\partial \mathbf {n} }{\partial t}}\right|^{2}-W(\mathbf {n} ,\nabla \mathbf {n} )-{\frac {\lambda }{2}}(1-|\mathbf {n} |^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5e06359a188e4c68e84e65b5d8e20d70405d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.038ex; height:6.343ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left|{\frac {\partial \mathbf {n} }{\partial t}}\right|^{2}-W(\mathbf {n} ,\nabla \mathbf {n} )-{\frac {\lambda }{2}}(1-|\mathbf {n} |^{2}),}"></span></dd></dl> <p>donde <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> es uno de los <a href="/wiki/Multiplicadores_de_Lagrange" title="Multiplicadores de Lagrange">multiplicadores de Lagrange</a> correspondiente a la restricción |<b>n</b>|=1. Limitaron su atención a las «ondas de separación», donde el campo del director toma la forma especial siguiente: </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 \mathbf {n} (x,y,z,t)=(\cos \varphi (x,t),\sin \varphi (x,t),0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} (x,y,z,t)=(\cos \varphi (x,t),\sin \varphi (x,t),0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5ea0d93cc1732b8ba4d8aa08a0a273f33770fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.401ex; height:2.843ex;" alt="{\displaystyle \mathbf {n} (x,y,z,t)=(\cos \varphi (x,t),\sin \varphi (x,t),0).}"></span></dd></dl> <p>Esta suposición reduce el lagrangiano 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}}\left(\varphi _{t}^{2}-a^{2}(\varphi )\varphi _{x}^{2}\right),\qquad a(\varphi ):={\sqrt {\alpha \sin ^{2}\varphi +\gamma \cos ^{2}\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> <mrow> <mo>(</mo> <mrow> <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> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>a</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>γ</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\varphi _{t}^{2}-a^{2}(\varphi )\varphi _{x}^{2}\right),\qquad a(\varphi ):={\sqrt {\alpha \sin ^{2}\varphi +\gamma \cos ^{2}\varphi }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc810639e80450146c98120928ce88a3ce22cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.534ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\varphi _{t}^{2}-a^{2}(\varphi )\varphi _{x}^{2}\right),\qquad a(\varphi ):={\sqrt {\alpha \sin ^{2}\varphi +\gamma \cos ^{2}\varphi }},}"></span></dd></dl> <p>y entonces la ecuación de Euler-Lagrange para el ángulo φ se convierte en: </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}=a(\varphi )[a(\varphi )\varphi _{x}]_{x}.}"> <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> <mi>a</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{tt}=a(\varphi )[a(\varphi )\varphi _{x}]_{x}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46455ff2aa8862a16bef5ac08ff8d5975ff5f3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.962ex; height:2.843ex;" alt="{\displaystyle \varphi _{tt}=a(\varphi )[a(\varphi )\varphi _{x}]_{x}.}"></span></dd></dl> <p>Hay soluciones constantes triviales φ=φ<sub>0</sub> correspondientes a los estados donde las moléculas en el cristal líquido están perfectamente alineados. La linealización alrededor de tal equilibrio conduce a la <a href="/wiki/Ecuaci%C3%B3n_de_onda" title="Ecuación de onda">ecuación de onda</a> lineal que permite la propagación de las ondas en ambas direcciones con velocidad </p> <dl><dt><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_{0}:=a(\varphi _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}:=a(\varphi _{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64b8569b8aab1b987d0e3ade9e59ab4599f68aaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.643ex; height:2.843ex;" alt="{\displaystyle a_{0}:=a(\varphi _{0})}"></span>,</dt></dl> <p>por lo que se puede esperar que la ecuación no lineal se comporte de manera similar. Con el fin de estudiar las ondas de movimiento correcto para las grandes <i>t</i>, se buscan soluciones asintóticas de la forma </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 (x,t;\epsilon )=\varphi _{0}+\epsilon \varphi _{1}(\theta ,\tau )+O(\epsilon ^{2}),}"> <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>;</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ϵ</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,t;\epsilon )=\varphi _{0}+\epsilon \varphi _{1}(\theta ,\tau )+O(\epsilon ^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d25dd28defa91d7c0092cfc7394a65d5379d10d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.746ex; height:3.176ex;" alt="{\displaystyle \varphi (x,t;\epsilon )=\varphi _{0}+\epsilon \varphi _{1}(\theta ,\tau )+O(\epsilon ^{2}),}"></span></dd></dl> <p>donde </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 \theta :=x-a_{0}t,\qquad \tau :=\epsilon t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>:=</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>,</mo> <mspace width="2em" /> <mi>τ</mi> <mo>:=</mo> <mi>ϵ</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta :=x-a_{0}t,\qquad \tau :=\epsilon t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2379dfe935855219f1ce5eef3770b76328e0f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.186ex; height:2.509ex;" alt="{\displaystyle \theta :=x-a_{0}t,\qquad \tau :=\epsilon t.}"></span></dd></dl> <p>Insertando esto en la ecuación, se encuentra en el orden <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 \epsilon ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8743fa5384acf4775698331a76dba47aed53ab29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.998ex; height:2.676ex;" alt="{\displaystyle \epsilon ^{2}}"></span> </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 _{1\tau }+a'(\varphi _{0})\varphi _{1}\varphi _{1\theta })_{\theta }={\frac {1}{2}}a'(\varphi _{0})\varphi _{1\theta }^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>τ</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>θ</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi _{1\tau }+a'(\varphi _{0})\varphi _{1}\varphi _{1\theta })_{\theta }={\frac {1}{2}}a'(\varphi _{0})\varphi _{1\theta }^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/172a3c97a3584c5843f5eb3f64ee8116cd0a9a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.683ex; height:5.176ex;" alt="{\displaystyle (\varphi _{1\tau }+a'(\varphi _{0})\varphi _{1}\varphi _{1\theta })_{\theta }={\frac {1}{2}}a'(\varphi _{0})\varphi _{1\theta }^{2}.}"></span></dd></dl> <p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Generalización"><span id="Generalizaci.C3.B3n"></span>Generalización</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecuaci%C3%B3n_de_Hunter-Saxton&action=edit&section=3" title="Editar sección: Generalización"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El análisis fue posteriormente generalizado por Alì y Hunter,<sup id="cite_ref-AH06_3-0" class="reference separada"><a href="#cite_note-AH06-3"><span class="corchete-llamada">[</span>3<span class="corchete-llamada">]</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 \mathbf {n} (x,y,z,t)=(\cos \varphi (x,t),\sin \varphi (x,t)\cos \psi (x,t),\sin \varphi (x,t)\sin \psi (x,t)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} (x,y,z,t)=(\cos \varphi (x,t),\sin \varphi (x,t)\cos \psi (x,t),\sin \varphi (x,t)\sin \psi (x,t)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d38ce7007eca009d34e68fa7d9e39c015b04784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.58ex; height:2.843ex;" alt="{\displaystyle \mathbf {n} (x,y,z,t)=(\cos \varphi (x,t),\sin \varphi (x,t)\cos \psi (x,t),\sin \varphi (x,t)\sin \psi (x,t)).}"></span></dd></dl> <p>Entonces, el lagrangiano es: </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}}\left(\varphi _{t}^{2}-a^{2}(\varphi )\varphi _{x}^{2}+\sin ^{2}\varphi \left[\psi _{t}^{2}-b^{2}(\varphi )\psi _{x}^{2}\right]\right),\qquad a(\varphi ):={\sqrt {\alpha \sin ^{2}\varphi +\gamma \cos ^{2}\varphi }},\quad b(\varphi ):={\sqrt {\beta \sin ^{2}\varphi +\gamma \cos ^{2}\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> <mrow> <mo>(</mo> <mrow> <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> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</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>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>a</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>γ</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </msqrt> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>γ</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\varphi _{t}^{2}-a^{2}(\varphi )\varphi _{x}^{2}+\sin ^{2}\varphi \left[\psi _{t}^{2}-b^{2}(\varphi )\psi _{x}^{2}\right]\right),\qquad a(\varphi ):={\sqrt {\alpha \sin ^{2}\varphi +\gamma \cos ^{2}\varphi }},\quad b(\varphi ):={\sqrt {\beta \sin ^{2}\varphi +\gamma \cos ^{2}\varphi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641ad9ea01a6a831bf65d185b01ebc499b992916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:114.844ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\varphi _{t}^{2}-a^{2}(\varphi )\varphi _{x}^{2}+\sin ^{2}\varphi \left[\psi _{t}^{2}-b^{2}(\varphi )\psi _{x}^{2}\right]\right),\qquad a(\varphi ):={\sqrt {\alpha \sin ^{2}\varphi +\gamma \cos ^{2}\varphi }},\quad b(\varphi ):={\sqrt {\beta \sin ^{2}\varphi +\gamma \cos ^{2}\varphi }}.}"></span></dd></dl> <p>Las ecuaciones correspondientes de Euler-Lagrange son ecuaciones de <a href="/wiki/Ondas_no_lineales" title="Ondas no lineales">ondas no lineales</a> acopladas para los ángulos φ y ψ, correspondiendo φ a ondas separadas y ψ a ondas giratorias. El caso anterior de Hunter-Saxton (ondas separadas puras) se recupera tomando la constante ψ, pero también se pueden considerar ondas separadas y giratorias acopladas en las que varían tanto φ como ψ. Las expansiones asintóticas similares a la anterior conducen a un sistema de ecuaciones que, después de renombrar y reescalar las variables, adopta la forma </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 (v_{t}+uv_{x})_{x}=0,\qquad u_{xx}=v_{x}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{t}+uv_{x})_{x}=0,\qquad u_{xx}=v_{x}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24a57c4409b71523089c5a29b11fec089340af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.833ex; height:3.009ex;" alt="{\displaystyle (v_{t}+uv_{x})_{x}=0,\qquad u_{xx}=v_{x}^{2},}"></span></dd></dl> <p>donde «u» se relaciona con φ y «v» con ψ. Este sistema implica, diferenciando la segunda ecuación con respecto a <i>t</i>, sustituyendo <i>v</i><sub><i>xt</i></sub>' de la primera ecuación, y eliminando <i>v</i> usando la segunda ecuación, que «u» satisface la siguiente ecuación: </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 \left[(u_{t}+uu_{x})_{x}-{\frac {1}{2}}\,u_{x}^{2}\right]_{x}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[(u_{t}+uu_{x})_{x}-{\frac {1}{2}}\,u_{x}^{2}\right]_{x}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37ce1a6e1b87d44cc2c91855ac941a57b27a2b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.073ex; height:6.176ex;" alt="{\displaystyle \left[(u_{t}+uu_{x})_{x}-{\frac {1}{2}}\,u_{x}^{2}\right]_{x}=0,}"></span></dd></dl> <p>así que (de manera bastante notable) la ecuación de Hunter-Saxton surge en este contexto también, pero de una manera diferente. </p> <div class="mw-heading mw-heading2"><h2 id="Estructuras_variables_e_integrabilidad">Estructuras variables e integrabilidad</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecuaci%C3%B3n_de_Hunter-Saxton&action=edit&section=4" title="Editar sección: Estructuras variables e integrabilidad"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Sistema_hamiltoniano_integrable" title="Sistema hamiltoniano integrable"> integrabilidad</a> de la ecuación de Hunter-Saxton o, más precisamente, la de su derivada </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_{t}+uu_{x})_{xx}=u_{x}u_{xx},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{t}+uu_{x})_{xx}=u_{x}u_{xx},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae55b6da31f2d2d2b3e4bee4566a2565935cbbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.44ex; height:2.843ex;" alt="{\displaystyle (u_{t}+uu_{x})_{xx}=u_{x}u_{xx},}"></span></dd></dl> <p>fue demostrado por Hunter y Zheng,<sup id="cite_ref-HZ94_4-0" class="reference separada"><a href="#cite_note-HZ94-4"><span class="corchete-llamada">[</span>4<span class="corchete-llamada">]</span></a></sup> quienes explicaron que esta ecuación se obtiene de la <a href="/wiki/Ecuaci%C3%B3n_de_Camassa-Holm" title="Ecuación de Camassa-Holm">ecuación de Camassa-Holm</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 u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>3</mn> <mi>u</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad48e5a072cd486ee55987586887e6c1a60a2e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.625ex; height:2.509ex;" alt="{\displaystyle u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}}"></span></dd></dl> <p>en el "límite de alta frecuencia" </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 (x,t)\mapsto (\epsilon x,\epsilon t),\qquad \epsilon \to 0.}"> <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> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mi>x</mi> <mo>,</mo> <mi>ϵ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,t)\mapsto (\epsilon x,\epsilon t),\qquad \epsilon \to 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf455f6b4461980d60c7db59e0051849b6abd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.574ex; height:2.843ex;" alt="{\displaystyle (x,t)\mapsto (\epsilon x,\epsilon t),\qquad \epsilon \to 0.}"></span></dd></dl> <p>Aplicando este procedimiento limitativo a un lagrangiano para la ecuación de Camassa-Holm, obtuvieron un lagrangiano </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}}_{2}={\frac {1}{2}}u_{x}^{2}+w(v_{t}+uv_{x})}"> <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"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>w</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{2}={\frac {1}{2}}u_{x}^{2}+w(v_{t}+uv_{x})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51a64bdb8f06602bcfe2d3b37708bfd84e27c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.995ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}_{2}={\frac {1}{2}}u_{x}^{2}+w(v_{t}+uv_{x})}"></span></dd></dl> <p>que genera la ecuación de Hunter-Saxton después de eliminar la <i>v</i> y la <i>w</i> de las ecuaciones de Euler-Lagrange para la <i>u</i>, la <i>v</i> y la <i>w</i>. Ya que también existe el más obvio lagrangiano </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}}_{1}=u_{x}u_{t}+uu_{x}^{2},}"> <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"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{1}=u_{x}u_{t}+uu_{x}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55266d3fdf4e9ad7fef4841eacccfa90536c03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.733ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{1}=u_{x}u_{t}+uu_{x}^{2},}"></span></dd></dl> <p>el Hunter-Saxton tiene dos estructuras variantes no equivalentes. Hunter y Zheng también obtuvieron una formulación bihamiltoniana y un <a href="/w/index.php?title=Par_Lax&action=edit&redlink=1" class="new" title="Par Lax (aún no redactado)">par Lax</a> de las estructuras correspondientes para la ecuación de Camassa-Holm de manera similar. </p><p>El hecho de que la ecuación Hunter-Saxton surge físicamente de dos maneras diferentes, como se muestra más arriba, fue utilizada por Alì y Hunter<sup id="cite_ref-AH06_3-1" class="reference separada"><a href="#cite_note-AH06-3"><span class="corchete-llamada">[</span>3<span class="corchete-llamada">]</span></a></sup> para explicar por qué tiene esta estructura bivariante o bihamiltoniana. </p> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecuaci%C3%B3n_de_Hunter-Saxton&action=edit&section=5" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-HS91-1"><span class="mw-cite-backlink">↑ <a href="#cite_ref-HS91_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HS91_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Hunter & Saxton 1991</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">de Gennes & Prost 1994 (Ch. 3)</span> </li> <li id="cite_note-AH06-3"><span class="mw-cite-backlink">↑ <a href="#cite_ref-AH06_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-AH06_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Alì & Hunter 2006</span> </li> <li id="cite_note-HZ94-4"><span class="mw-cite-backlink"><a href="#cite_ref-HZ94_4-0">↑</a></span> <span class="reference-text">Hunter & Zheng 1994</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecuaci%C3%B3n_de_Hunter-Saxton&action=edit&section=6" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFAlìHunter2006" class="citation">Alì, Giuseppe; Hunter, John K. (2006), <i>Orientation waves in a director field with rotational inertia</i>, <small><a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/math.AP/0609189">math.AP/0609189</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.au=Al%C3%AC%2C+Giuseppe&rft.au=Hunter%2C+John+K.&rft.aufirst=Giuseppe&rft.aulast=Al%C3%AC&rft.btitle=Orientation+waves+in+a+director+field+with+rotational+inertia&rft.date=2006&rft.genre=book&rft_id=info%3Aarxiv%2Fmath.AP%2F0609189&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFde_GennesProst1994" class="citation">de Gennes, Pierre-Gilles; Prost, Jacques (1994), <i>The Physics of Liquid Crystals</i>, International Series of Monographs on Physics (2nd edición), Oxford University Press, <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/0-19-852024-7" title="Especial:FuentesDeLibros/0-19-852024-7">0-19-852024-7</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.au=Prost%2C+Jacques&rft.au=de+Gennes%2C+Pierre-Gilles&rft.aufirst=Pierre-Gilles&rft.aulast=de+Gennes&rft.btitle=The+Physics+of+Liquid+Crystals&rft.date=1994&rft.edition=2nd&rft.genre=book&rft.isbn=0-19-852024-7&rft.pub=Oxford+University+Press&rft.series=International+Series+of+Monographs+on+Physics&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFHunterSaxton1991" class="citation">Hunter, John K.; Saxton, Ralph (1991), «Dynamics of director fields», <i>SIAM J. Appl. Math.</i> <b>51</b> (6): 1498-1521, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1137%2F0151075">10.1137/0151075</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=Dynamics+of+director+fields&rft.au=Hunter%2C+John+K.&rft.au=Saxton%2C+Ralph&rft.aufirst=John+K.&rft.aulast=Hunter&rft.date=1991&rft.genre=article&rft.issue=6&rft.jtitle=SIAM+J.+Appl.+Math.&rft.pages=1498-1521&rft.volume=51&rft_id=info%3Adoi%2F10.1137%2F0151075&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFHunterZheng1994" class="citation">Hunter, John K.; Zheng, Yuxi (1994), «On a completely integrable nonlinear hyperbolic variational equation», <i>Physica D</i> <b>79</b> (2–4): 361-386, <small><a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1994PhyD...79..361H">1994PhyD...79..361H</a></small>, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2FS0167-2789%2805%2980015-6">10.1016/S0167-2789(05)80015-6</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=On+a+completely+integrable+nonlinear+hyperbolic+variational+equation&rft.au=Hunter%2C+John+K.&rft.au=Zheng%2C+Yuxi&rft.aufirst=John+K.&rft.aulast=Hunter&rft.date=1994&rft.genre=article&rft.issue=2%E2%80%934&rft.jtitle=Physica+D&rft.pages=361-386&rft.volume=79&rft_id=info%3Abibcode%2F1994PhyD...79..361H&rft_id=info%3Adoi%2F10.1016%2FS0167-2789%2805%2980015-6&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFBealsSattingerSzmigielski2001" class="citation"><a href="/w/index.php?title=Richard_Beals_(mathematician)&action=edit&redlink=1" class="new" title="Richard Beals (mathematician) (aún no redactado)">Beals, Richard</a>; Sattinger, David H.; Szmigielski, Jacek (2001), <a rel="nofollow" class="external text" href="http://www.math.usu.edu/~dhs/hunter209.ps">«Inverse scattering solutions of the Hunter–Saxton equation»</a>, <i>Applicable Analysis</i> <b>78</b> (3–4): 255-269, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F00036810108840938">10.1080/00036810108840938</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=Inverse+scattering+solutions+of+the+Hunter%E2%80%93Saxton+equation&rft.au=Beals%2C+Richard&rft.au=Sattinger%2C+David+H.&rft.au=Szmigielski%2C+Jacek&rft.aufirst=Richard&rft.aulast=Beals&rft.date=2001&rft.genre=article&rft.issue=3%E2%80%934&rft.jtitle=Applicable+Analysis&rft.pages=255-269&rft.volume=78&rft_id=http%3A%2F%2Fwww.math.usu.edu%2F~dhs%2Fhunter209.ps&rft_id=info%3Adoi%2F10.1080%2F00036810108840938&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFBressanConstantin2005" class="citation">Bressan, Alberto; Constantin, Adrian (2005), «Global solutions of the Hunter–Saxton equation», <i>SIAM J. Math. Anal.</i> <b>37</b> (3): 996-1026, <small><a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/math/0502059">math/0502059</a></small>, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1137%2F050623036">10.1137/050623036</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=Global+solutions+of+the+Hunter%E2%80%93Saxton+equation&rft.au=Bressan%2C+Alberto&rft.au=Constantin%2C+Adrian&rft.aufirst=Alberto&rft.aulast=Bressan&rft.date=2005&rft.genre=article&rft.issue=3&rft.jtitle=SIAM+J.+Math.+Anal.&rft.pages=996-1026&rft.volume=37&rft_id=info%3Aarxiv%2Fmath%2F0502059&rft_id=info%3Adoi%2F10.1137%2F050623036&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFHoldenKarlsenRisebro2007" class="citation"><a href="/w/index.php?title=Helge_Holden&action=edit&redlink=1" class="new" title="Helge Holden (aún no redactado)">Holden, Helge</a>; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070922012502/http://www.math.uio.no/eprint/pure_math/2005/20-05/index.html">«Convergent difference schemes for the Hunter–Saxton equation»</a>, <i>Math. Comp.</i> <b>76</b> (258): 699-745, <small><a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2007MaCom..76..699H">2007MaCom..76..699H</a></small>, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0025-5718-07-01919-9">10.1090/S0025-5718-07-01919-9</a></small>, archivado desde <a rel="nofollow" class="external text" href="http://www.math.uio.no/eprint/pure_math/2005/20-05/index.html">el original</a> el 22 de septiembre de 2007<span class="reference-accessdate">, consultado el 12 de julio de 2020</span></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=Convergent+difference+schemes+for+the+Hunter%E2%80%93Saxton+equation&rft.au=Holden%2C+Helge&rft.au=Karlsen%2C+Kenneth+Hvistendahl&rft.au=Risebro%2C+Nils+Henrik&rft.aufirst=Helge&rft.aulast=Holden&rft.date=2007&rft.genre=article&rft.issue=258&rft.jtitle=Math.+Comp.&rft.pages=699-745&rft.volume=76&rft_id=http%3A%2F%2Fwww.math.uio.no%2Feprint%2Fpure_math%2F2005%2F20-05%2Findex.html&rft_id=info%3Abibcode%2F2007MaCom..76..699H&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-07-01919-9&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFHunterZheng1995" class="citation">Hunter, John K.; Zheng, Yuxi (1995), «On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions», <i>Arch. Rational Mech. Anal.</i> <b>129</b> (4): 305-353, <small><a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1995ArRMA.129..305H">1995ArRMA.129..305H</a></small>, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF00379259">10.1007/BF00379259</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=On+a+nonlinear+hyperbolic+variational+equation.+I.+Global+existence+of+weak+solutions&rft.au=Hunter%2C+John+K.&rft.au=Zheng%2C+Yuxi&rft.aufirst=John+K.&rft.aulast=Hunter&rft.date=1995&rft.genre=article&rft.issue=4&rft.jtitle=Arch.+Rational+Mech.+Anal.&rft.pages=305-353&rft.volume=129&rft_id=info%3Abibcode%2F1995ArRMA.129..305H&rft_id=info%3Adoi%2F10.1007%2FBF00379259&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFHunterZheng1995" class="citation">Hunter, John K.; Zheng, Yuxi (1995), «On a nonlinear hyperbolic variational equation. II. The zero-viscosity and dispersion limits», <i>Arch. Rational Mech. Anal.</i> <b>129</b> (4): 355-383, <small><a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1995ArRMA.129..355H">1995ArRMA.129..355H</a></small>, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF00379260">10.1007/BF00379260</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=On+a+nonlinear+hyperbolic+variational+equation.+II.+The+zero-viscosity+and+dispersion+limits&rft.au=Hunter%2C+John+K.&rft.au=Zheng%2C+Yuxi&rft.aufirst=John+K.&rft.aulast=Hunter&rft.date=1995&rft.genre=article&rft.issue=4&rft.jtitle=Arch.+Rational+Mech.+Anal.&rft.pages=355-383&rft.volume=129&rft_id=info%3Abibcode%2F1995ArRMA.129..355H&rft_id=info%3Adoi%2F10.1007%2FBF00379260&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFLenells2007" class="citation">Lenells, Jonatan (2007), «The Hunter–Saxton equation describes the geodesic flow on a sphere», <i>J. Geom. Phys.</i> <b>57</b> (10): 2049-2064, <small><a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2007JGP....57.2049L">2007JGP....57.2049L</a></small>, <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2Fj.geomphys.2007.05.003">10.1016/j.geomphys.2007.05.003</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEcuaci%C3%B3n+de+Hunter-Saxton&rft.atitle=The+Hunter%E2%80%93Saxton+equation+describes+the+geodesic+flow+on+a+sphere&rft.au=Lenells%2C+Jonatan&rft.aufirst=Jonatan&rft.aulast=Lenells&rft.date=2007&rft.genre=article&rft.issue=10&rft.jtitle=J.+Geom.+Phys.&rft.pages=2049-2064&rft.volume=57&rft_id=info%3Abibcode%2F2007JGP....57.2049L&rft_id=info%3Adoi%2F10.1016%2Fj.geomphys.2007.05.003&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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