Document Zbl 1234.35006

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Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898717-05-1/pbk; 978-0-89871-968-0/ebook). xxvi, 430 p. (2010). </div> <div class="abstract">The aim of the monograph is to prepare graduate students of applied mathematics for the mathematical treatment of nonlinear wave phenomena, but it is also useful for scientists in other sciences, especially in hydromechanics, optics, quantum theory, statistical mechanics and engineering. The 430-pages-long textbook comprises studies for integrable and nonintegrable nonlinear wave equations. The theory and experiments are treated, and analytical and numerical methods for the characterization and solution of the equations are developed. If a problem can be solved by various methods, the author explains the reason for his choices of the presented technique. With few exceptions the monograph is self-contained. Calculations are carefully explained. Readers without skills in physics who are interested in the physical background of the equations may need additional textbooks.<br class="zbmathjax-paragraph">The book consists of 7 chapters divided into sections and partly subsections, a comprehensive bibliography, and an index with the most important notions. Particularly, numerous bibliographical data of the original publications to the treated subjects are included in the sections. The text is completed by a number of about hundred figures and computer graphics. <span class="zbmathjax-texttt">Matlab</span> codes are presented for all treated computational methods. The codes are also available on an associated web page and they can be easily adapted to own corresponding problems of the users. Exercises to extend the skill of the readers are not formulated.<br class="zbmathjax-paragraph">In the preface the author starts with well-known linear wave phenomena, such as low-amplitude water waves and low-intensity light beams which disappear after a short time by dispersion or diffraction, respectively. Then he gives a short historical introduction into the development of nonlinear wave research starting with a citation of Russel’s observation of a water wave which did not disperse and that could not be explained by linear theory. The first mathematical formulation of this problem was given by Korteweg and de Vries in 1895 taking into account nonlinear effects. But an expanded research on nonlinear waves sets in the 1960s. Other nonlinear wave equations, such as nonlinear Schrödinger equations, sine-Gordon equations, and Kadomtsev-Petviashvili equations including analytical and numerical solution methods, are of great interest until today. Nonlinear optics and Bose-Einstein condensates have become a driving force since the 1980s and 1995, respectively.<br class="zbmathjax-paragraph">Chapter 1 is devoted to the derivation of nonlinear wave equations from known physical systems under appropriate asymptotic limits. First, nonlinear Schrödinger (NLS) equations describing the evolution of the envelope of low-amplitude slowly varying wave packets are deduced from the Korteweg-de Vries (KdV) and the modified KdV equation taking into account only the quadratic and cubic nonlinearities for different dispersion relations. Then, starting from Maxwell’s equations, generalized NLS equations are derived for the light beam propagation, especially in a homogeneous Kerr nonlinear medium and a non Kerr nonlinear medium including the case when the light is confined in one direction by a waveguide and covering the treatment of photonic crystal fibers and laser-written waveguides with the corresponding refraction index.<br class="zbmathjax-paragraph">Other topics of this chapter are the formulation of NLS equations with higher-order corrections (such as higher-order corrections to the NLS equation for optical pulse transmission in fiber optics), of coupled NLS equations, and of a NLS equation with an external potential, the so-called Gross-Pitaevskii equation, describing the state of a Bose-Einstein condensate by a collective wave function.<br class="zbmathjax-paragraph">While the first chapter provides information about the origins of nonlinear wave equations, all the following chapters are devoted to the characterization and solution of these equations. The next three chapters deal with the integrable theory for nonlinear wave equations. Solitary waves of the above-mentioned KdV equations, NLS equations, sine-Gordon equations, Kadomtsev-Petviashvili equations, and other equations, which do not change their speed or shape if they pass through each other, can be exactly computed by the inverse scattering transform method. For weakly perturbed integrable equations, soliton perturbation theory is introduced.<br class="zbmathjax-paragraph">If a nonlinear equation is the compatibility condition between a spatial and a temporal linear operator (the so-called Lax pair of the nonlinear equation), then this equation is integrable by the inverse scattering transform method. Based on this idea, the method is developed for an initial value problem of the NLS equation in Chapter 2. The spatial equation, the so-called Zakharov-Shabat scattering problem, gives the scattering data at a fixed time $t_0$, from which the scattering data can be computed at any later time $t$. The solution $u(x,t)$ of the NLS equation can then be reconstructed from the scattering data (thus inverse scattering). The author compares this method with the solution of partial differential equations by the Fourier transform and the inverse Fourier transform. The scattering data are the analog of the Fourier coefficients. For the implementation of the Zakharov-Shabat system the author prefers a technique that results in a regular or, in a more general case, in a nonregular Riemann-Hilbert problem. The unique solution of the regular matrix Riemann-Hilbert problem in the complex plane under canonical normalization condition is proved, and the solution can be given in terms of a Fredholm integral equation, from which the long-time asymptotic state of the NLS solution, consisting of a radiation and a nonradiative part (the soliton component), can be computed. The radiation part will disperse over the time, that is, the soliton component will dominate the long-time behavior. A theorem is formulated and proved that reduces the nonregular problem to a regular one giving the minimal scattering data from which the time evolution and finally the solution $u(x,t)$ can be obtained. The mentioned nonradiative part is called an $N$-soliton solution. The solution is also called a reflectionless potential. The cases $N = 1$ and $N = 2$ are discussed in detail and well presented by graphics.<br class="zbmathjax-paragraph">Other topics of this chapter are the derivation of conservation laws of the NLS equation and the determination of discrete eigenvalues of the Zakharov-Shabat system for various initial value conditions, such as the Satsuma-Yajima initial condition, the box initial condition, and conditions that are extended from these two conditions. The eigenvalues correspond with the discrete scattering data. The Zakharov-Shabat system can be rewritten as an eigenvalue problem which generally has to be solved by a numerical method. Two methods are proposed by the author: (i) the approximation of the spatial derivatives by a finite-difference scheme and the solution of the corresponding matrix eigenvalue problem by the Arnoldi algorithm, (ii) the use of a Fourier collocation method turning the Zakharov-Shabat system into a matrix eigenvalue problem for the Fourier coefficients or first taking the Fourier transform of the Zakharov-Shabat system and then discretizing the transformed equations in the Fourier space, and the solution of the corresponding $N$-dimensional eigenvalue problem by the QR algorithm (for small $N$) or by the Arnoldi method that gives a few eigenvalues. A <span class="zbmathjax-texttt">Matlab</span> code of the last method is displayed. The author reports that the first method can give spurious modes. A reason for it is not outlined.<br class="zbmathjax-paragraph">The next topics of Chapter 2 are the proof of the completeness of eigenfunctions and the concept of squared eigenfunctions for the Zakharov-Shabat system which are important for the studies of perturbed integrable equations treated in Chapter 4.<br class="zbmathjax-paragraph">As found by Ablowitz, Kaup, Newell and Sigur (AKNS), an infinite hierarchy of integrable equations can be constructed with each scattering operator. The AKNS hierarchy is derived. For the Zakharov-Shabat system this hierarchy contains the NLS equation as special case, but also the KdV, the modified KdV, the complex modified KdV equation, and coupled NLS equations. Unusual soliton dynamics, such as changing the shape or amplitude during propagation or blow up to infinity at finite time, are treated.<br class="zbmathjax-paragraph">While the last chapter deals with the AKNS hierarchy associated with the second-order Zakharov-Shabat scattering problem, Chapter 3 is concerned with a larger class of integrable equations with higher-order scattering problems, such as the vector NLS systems. The problems are with few exceptions similar to those in Chapter 2 and give a deeper insight into the treatment of integrable equations. Especially, the user will be familiar with the Manakov solitons, the coupled focusing-defocusing NLS system, and the solutions of the Sasa-Satsuma equation.<br class="zbmathjax-paragraph">While in the last two chapters certain nonlinear physical wave systems at the lowest order are described, Chapter 4 is devoted to NLS equations including perturbation terms, such as damping, higher-order nonlinearity, and higher-order dispersion. The shapes of the corresponding solitons now may be distorted over time, and the energy radiation may impact the evolution of the solitons. Thus, a soliton perturbation theory is introduced for this subject. The author has opted for the direct perturbation theory for NLS equations due to its simplicity and many-sidedness among numerous different perturbation theories for the presentation in this book. The direct perturbation theory is substantially based on a multiscale perturbation expansion, containing a slow (for the soliton parameters) and a fast (for the energy radiation) time scaling, and the direct solving of corresponding linearized wave equations using squared eigenfunctions, already introduced in Chapter 2.<br class="zbmathjax-paragraph">If a light pulse in an optical fiber is very short, heigh-order effects, such as the Raman effect, the self-steepening effect, and the third-order dispersion effect, have to be taken into account in the NLS equation. The carefully deduced direct perturbation theory is applied to the corresponding equations. Other topics of this chapter are the evolution of a perturbed soliton in the integrable NLS equation if the initial condition is a perturbed single soliton, and of weak interactions of NLS solitons if the initial condition is formed by two separated single solitons. The last section of Chapter 4 is devoted to the application of the soliton perturbation theory to the complex modified KdV equation, which exhibits a number of new features in comparison to the NLS perturbation theory.<br class="zbmathjax-paragraph">Most nonlinear wave equations are nonintegrable, that is, they cannot be solved by inverse scattering theory or soliton perturbation theory. Analytical and numerical methods of this forty-year-old science for the solution and characterization of these nonlinear phenomena are the subject of the last three chapters. Solitary waves in nonintegrable equations exhibit a significant different behavior, such as instability and collisions, in comparison to integrable systems.<br class="zbmathjax-paragraph">Chapter 5 starts with a generalized NLS equation with three cases of a dual-power nonlinearity, which describes nonlinear light propagation in a non-Kerr medium. The consideration of the corresponding solitary waves under small perturbations leads to the linearization spectrum of these waves, which can be used to characterize their stability and other properties. It is derived that the spectrum, in the case where the dual-power type gives an integrable NLS equation, does not contain nonzero eigenvalues, but in the other two cases it contains a pair of discrete eigenvalues on the imaginary axis (internal modes) or a pair of real eigenvalues. This appearance of nonzero eigenvalues is a signature of nonintegrable equations and plays a fundamental role in the dynamics of the solitary waves. The solitary waves are unstable if the spectrum contains any eigenvalue with positive real part. The origin of the unstable eigenvalue is associated with the slope of the power curve of the solitary wave. The eigenvalues depend on a propagation constant of the solitary wave. Properly changing this constant, the pair of real eigenvalues moves to the origin, collides on the power minimum point, and then bifurcates along the imaginary axis. This interrelation gives a stability criterion, the so-called Vakhitov-Kolokolov stability criterion, that is generalized and proved.<br class="zbmathjax-paragraph">Other topics of this comprehensive chapter are the exponential asymptotic technique for nonlocal waves, embedded solitons, fractal scattering phenomena in collisions of solitary waves (which are important in soliton-based fiber communication systems, where pulses traveling in different frequency channels collide), the transverse instability of solitary waves, and the wave collapse in two-dimensional NLS equations.<br class="zbmathjax-paragraph">All coefficients of the equations in the previous chapters are space-independent, which means that the corresponding media are assumed to be homogeneous. In Chapter 6 the nonlinear wave equations are treated for periodic media, that is, the equations have spatially periodic coefficients. Periodic media considerably change the dispersion relations, which causes numerous new nonlinear phenomena in nonlinear optics (photonic crystal fibers, etched waveguide arrays on AlGaAs, laser-written waveguide arrays), and Bose-Einstein condensates (loading into an optical lattice that forms a periodic potential for the condensate), mainly studied in the last decade. Focusing on the Kerr model, the reader is confronted with Bloch modes and bandgaps, their envelope equations, gap solitons, and the analytical calculation of the eigenvalue bifurcation near band edges. Experimental results performed in photorefractive crystals, waveguide arrays, and Bose-Einstein condensates are presented in the last section of this chapter. It is reported that the theories coincide with the experimental results.<br class="zbmathjax-paragraph">The last chapter is devoted to numerical methods for nonlinear wave equations, which are especially important for nonintegrable equations. Numerical methods have been already included into Chapters 5 and 6 because they are closely connected with the analytical investigations, but in this chapter all these and other methods are described including accuracy, numerical stability, convergence rates, and <span class="zbmathjax-texttt">Matlab</span> codes. Numerical methods for evolution simulations are the subject of the first section. The pseudospectral method consist of the steps: discretizing the initial value problem in space, using a discrete Fourier transform to evaluate the spatial derivative, and advancing in time by for instance a Runge-Kutta method. Investigated are also another spectral method, the integrating-factor method, and the split-step method.<br class="zbmathjax-paragraph">Numerical methods for solitary waves are considered in Section 7.2. Especially, numerical algorithms for unstable solitary waves are of interest. The treated Petviashvili-type methods are based on a fixed-point iteration. They converge fast, but do so only in the ground states of the nonlinear wave equations. Another method is the accelerated imaginary-time evolution algorithm, which uses a preconditioning technique and is connected to the linear stability of the solitary wave. The method converges similarly to the Petviashvili method only for the ground states and diverges for exited states. Two further iteration methods are described, the squared operator iteration algorithm (SOM) and the Newton conjugate-gradient method (Newton-CG method), which converge for both the ground states and the excited states. SOM contains a preconditioning technique that accelerates the convergence, but the squaring of the operator squares also the condition number which leads to a slower convergence. Thus, in addition to the SOM, a modified SOM is developed that overcomes this problem and converges faster. The preconditioned CG algorithm forms the inner iteration of the Newton-CG method which converges faster than the other methods.<br class="zbmathjax-paragraph">The linear-stability spectrum of a solitary wave consists of the eigenvalues of the wave. The spectrum characterizes the solitons. Thus, the last section describes numerical algorithms to solve the linear-stability eigenvalue problem. Two methods are presented: the Fourier collocation method for the whole spectrum and the Newton conjugate-gradient method for individual eigenvalues.<br class="zbmathjax-paragraph">Even though the monograph communicates a substantial amount of knowledge on nonlinear wave research, a number of results in this field could not be included, such as (mentioned by the author) the Hirota method, the Darboux and Bäcklund transformations, Painlevé properties, and nonlinear water waves.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10878">Georg Hebermehl (Berlin)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5816516">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5816516">Cited in <strong>462</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35-02" title="MSC2020">35-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37-02" title="MSC2020">37-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65-02" title="MSC2020">65-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to numerical analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q55" title="MSC2020">35Q55</a> </td> <td class="space"> NLS equations (nonlinear Schrödinger equations) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q41" title="MSC2020">35Q41</a> </td> <td class="space"> Time-dependent Schrödinger equations and Dirac equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q51" title="MSC2020">35Q51</a> </td> <td class="space"> Soliton equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J10" title="MSC2020">35J10</a> </td> <td class="space"> Schrödinger operator, Schrödinger equation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35P10" title="MSC2020">35P10</a> </td> <td class="space"> Completeness of eigenfunctions and eigenfunction expansions in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37K10" title="MSC2020">37K10</a> </td> <td class="space"> Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37K15" title="MSC2020">37K15</a> </td> <td class="space"> Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N25" title="MSC2020">65N25</a> </td> <td class="space"> Numerical methods for eigenvalue problems for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N35" title="MSC2020">65N35</a> </td> <td class="space"> Spectral, collocation and related methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37K40" title="MSC2020">37K40</a> </td> <td class="space"> Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A78A60" title="MSC2020">78A60</a> </td> <td class="space"> Lasers, masers, optical bistability, nonlinear optics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A82C10" title="MSC2020">82C10</a> </td> <td class="space"> Quantum dynamics and nonequilibrium statistical mechanics (general) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlinear+waves">nonlinear waves</a>; <a href="/?q=ut%3Aintegrable+systems">integrable systems</a>; <a href="/?q=ut%3Anonintegrable+systems">nonintegrable systems</a>; <a href="/?q=ut%3Anonlinear+Schr%C3%B6dinger+equation">nonlinear Schrödinger equation</a>; <a href="/?q=ut%3Asoliton">soliton</a>; <a href="/?q=ut%3Ainverse+scattering+method">inverse scattering method</a>; <a href="/?q=ut%3ARiemann-Hilbert+formulation">Riemann-Hilbert formulation</a>; <a href="/?q=ut%3Aconservation+laws">conservation laws</a>; <a href="/?q=ut%3Asoliton+perturbation+theory">soliton perturbation theory</a>; <a href="/?q=ut%3ALax+pair">Lax pair</a>; <a href="/?q=ut%3AZakharov-Shabat+system">Zakharov-Shabat system</a>; <a href="/?q=ut%3AAKNS+hierarchy">AKNS hierarchy</a>; <a href="/?q=ut%3Asquared+eigenfunctions">squared eigenfunctions</a>; <a href="/?q=ut%3AManakov+system">Manakov system</a>; <a href="/?q=ut%3ASasa-Satsuma+equation">Sasa-Satsuma equation</a>; <a href="/?q=ut%3Ahigher-order+effects">higher-order effects</a>; <a href="/?q=ut%3AVakhitov-Kolokolov+stability+criterion">Vakhitov-Kolokolov stability criterion</a>; <a href="/?q=ut%3Aembedded+solitons">embedded solitons</a>; <a href="/?q=ut%3Afractal+scattering">fractal scattering</a>; <a href="/?q=ut%3Aperiodic+media">periodic media</a>; <a href="/?q=ut%3ABloch+waves">Bloch waves</a>; <a href="/?q=ut%3Abandgaps">bandgaps</a>; <a href="/?q=ut%3Aspectrum">spectrum</a>; <a href="/?q=ut%3Aeigenvalue+bifurcation">eigenvalue bifurcation</a>; <a href="/?q=ut%3Apseudospectral+method">pseudospectral method</a>; <a href="/?q=ut%3Asplit-step+method%2C+integrating+factor+method">split-step method, integrating factor method</a>; <a href="/?q=ut%3APetviashvili+method">Petviashvili method</a>; <a href="/?q=ut%3Asquared-operator+method">squared-operator method</a>; <a href="/?q=ut%3ANewton+conjugate-gradient+method">Newton conjugate-gradient method</a>; <a href="/?q=ut%3Aaccelerated+imaginary-time+evolution+method">accelerated imaginary-time evolution method</a>; <a href="/?q=ut%3Alinear+stability">linear stability</a>; <a href="/?q=ut%3AFourier+collocation+method">Fourier collocation method</a>; <a href="/?q=ut%3Amonograph">monograph</a>; <a href="/?q=ut%3Atextbook">textbook</a>; <a href="/?q=ut%3Asine-Gordon+equation">sine-Gordon equation</a>; <a href="/?q=ut%3AKadomtsev-Petviashvili+equation">Kadomtsev-Petviashvili equation</a>; <a href="/?q=ut%3ABose-Einstein+condensate">Bose-Einstein condensate</a>; <a href="/?q=ut%3AMaxwell%27s+equations">Maxwell’s equations</a>; <a href="/?q=ut%3AGross-Pitaevskii+equation">Gross-Pitaevskii equation</a>; <a href="/?q=ut%3AFredholm+integral+equation">Fredholm integral equation</a>; <a href="/?q=ut%3Afinite-difference+scheme">finite-difference scheme</a>; <a href="/?q=ut%3AArnoldi+algorithm">Arnoldi algorithm</a>; <a href="/?q=ut%3Aoptical+fiber">optical fiber</a>; <a href="/?q=ut%3Awaveguide">waveguide</a>; <a href="/?q=ut%3Adiscrete+Fourier+transform">discrete Fourier transform</a>; <a href="/?q=ut%3ARunge-Kutta+method">Runge-Kutta method</a>; <a href="/?q=ut%3Apreconditioning">preconditioning</a>; <a href="/?q=ut%3Acondition+number">condition number</a></div> <div class="software"> <h3>Software:</h3><a href="/software/558">Matlab</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1234.35006" data-ciurl="/ci/05816516" data-biburl="/bibtex/05816516.bib" data-amsurl="/amsrefs/05816516.bib" data-xmlurl="/xml/05816516.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05816516.pdf" title="Zbl 1234.35006 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1137/1.9780898719680" aria-label="DOI for “Nonlinear waves in integrable and nonintegrable systems.”" title="10.1137/1.9780898719680">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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