Nonlinear Waves in Integrable and Nonintegrable Systems [book]

Jianke Yang
2010 unpublished
430pp., ?? (US$ 85.00), ISBN 978-0-898717-05-1 (Society for Industrial and Applied Mathematics, Philadelphia, 2010). Modern nonlinear wave theory is a rapidly developing area which includes a large variety of sophisticated ideas and powerful methods, as well as a vast number of important real-life applications. The book by Jianke Yang (2010) entitled 'Nonlinear Waves in Integrable and Nonintegrable Systems' will help an interested reader to discover many diverse aspects of nonlinear wave theory
more » ... nlinear wave theory in a single book. The author is neither solely driven by the intrinsic beauty of the underlying mathematical theories, although he definitely admires them as he tells the story, nor by the efficiency of numerical methods, although he clearly appreciates them. It will be fair to say that the author is driven by the desire to develop all methods which can help us to understand, describe and predict physical phenomena, and this book is a reflection of the author's own research path 'from integrable to nonintegrable equations, from analysis to numerics, and from theory to experiments'. The physical phenomena to which this book is most relevant are nonlinear wave processes in optics and Bose-Einstein condensates, although the main equations considered in the book are of universal nature, and most of the approaches have counterparts relevant to many other applications. The first chapter begins by introducing a very important nonlinear wave model, the Nonlinear Schrödinger (NLS) equation. The NLS equation is first derived from a Korteweg -de Vries (KdV) -type equation using asymptotic multiple -scales expansions. This powerful technique, introduced here in a very simple and instructive example and used in some consequent chapters, allows one to obtain the mathematical 'DNA' of many unmanageable problems -their reduced mathematical models amenable to analysis. The author then discusses, with useful references, the NLS and generalised NLS equations describing the nonlinear light propagation in a Kerr and non-Kerr medium, as well as the NLS equation with higher-order corrections and the coupled NLS equations, in the context of nonlinear waves in optics, and the Gross-Pitaevsky equation, in the context of waves in Bose-Einstein condensates. All these models are, in a sense, close relatives of the basic integrable model, the NLS equation. However, their solutions, and methods needed in order to study them, are different. The second chapter is devoted to the integrability theory for the NLS equation by the Inverse Scattering Transform, the method which was first developed by Gardner, Green, Kruskal and Miura in 1967 for the KdV equation. The treatment of the NLS equation in the book is based on the Riemann -Hilbert formulation (Zakharov and Shabat, 1979; Zakharov, Manakov, Novikov, Pitaevskii, 1980). The integrability of the NLS equation was established by Zakharov and Shabat in 1971. Integrable models have many remarkable and beautiful mathematical properties, from the existence of Lax pairs, related integrable hierarchies, recursion operators and infinite numbers of conservation laws, to N -soliton solutions and connections between solutions of the linearised equations and squared eigenfunctions of the associated spectral problem, all of which are discussed in this chapter, in the context of the NLS equation and the AKNS hierarchy (Ablowitz, Kaup, Newell, and Segur (1974)). The purer aspects of integrable systems are left out of the scope of the book. The third chapter is an extension of the integrability theory for a single NLS equation, associated with the second order Zakharov -Shabat spectral problem, to integrable models associated with the higher-order spectral problems, such as the vector NLS system. The models belong to one and the same integrable hierarchy generalising the AKNS hierarchy, and are treated together using the Rieman -Hilbert formulation for the hierarchy. The author highlights the existing similarities and differences in the treatment of the scalar and vector problems. The chapter includes the discussion of multisoliton solutions of the Manakov system, the coupled focusing -defocusing NLS system and the Sasa-Satsuma equation, three integrable models having a third-order spectral problem. The fourth chapter deals with nonintegrable equations which constitute weak perturbations of integrable models. Such nearly -integrable equations naturally appear in the studies of various physical problems. In this chapter, soliton perturbation theory based on asymptotic multiple -scales expansions is developed for the perturbed NLS equation. The emerging linearised equations are solved using the squared eigenfunctions of the spectral problem associated with the unperturbed integrable equation. The perturbation theory is applied to study the evolution of the NLS soliton under the higher-order optical effects, including the Raman, selfsteepening and third-order dispersion effects. The chapter also includes the studies of weak interactions of the NLS solitons, and soliton perturbation theory for the complex modified KdV equation. In the second case, the solitary wave solutions have new features; generically a solitary wave looses energy due to radiation of the Fourier mode which is in resonance with the solitary wave. Importantly, the amplitude of this oscillating 'tail' does not decrease at the fast time scale. Remarkably, among these nonlocal solitary wave solutions there exists a family of truly localised solutions, called 'embedded solitons' (Yang, Malomed and Kaup (1999) ).
doi:10.1137/1.9780898719680 fatcat:5rnprlyj2zhena3de5hyhzhp2i