Swift progress and new applications characterize the area of solitons and the inverse scattering transform. There are rapid developments in current nonlinear optical technology: Larger intensities are more available; pulse widths are smaller; relaxation times and damping rates are less significant. In keeping with these advancements, exactly integrable soliton equations, such as $3$-wave resonant interactions and second harmonic generation, are becoming more and more relevant in experimental applications. Techniques are now being developed for using these interactions to frequency convert high intensity sources into frequency regimes where there are no lasers. Other experiments involve using these interactions to develop intense variable frequency sources, opening up even more possibilities. This volume contains new developments and state-of-the-art research arising from the conference on the ""Legacy of the Inverse Scattering Transform"" held at Mount Holyoke College (South Hadley, MA). Unique to this volume is the opening section, ""Reviews"". This part of the book provides reviews of major research results in the inverse scattering transform (IST), on the application of IST to classical problems in differential geometry, on algebraic and analytic aspects of soliton-type equations, on a new method for studying boundary value problems for integrable partial differential equations (PDEs) in two dimensions, on chaos in PDEs, on advances in multi-soliton complexes, and on a unified approach to integrable systems via Painleve analysis. This conference provided a forum for general exposition and discussion of recent developments in nonlinear waves and related areas with potential applications to other fields. The book will be of interest to graduate students and researchers interested in mathematics, physics, and engineering.
The legacy of the IST by D. J. Kaup Application of inverse scattering method to problems of differential geometry by V. Zakharov Algebraic and analytic aspects of soliton type equations by V. S. Gerdjikov Differential forms, spectral theory, and boundary value problems by A. S. Fokas Chaos in partial differential equations by Y. C. Li Multi-soliton complexes by N. N. Akhmediev, A. A. Sukhorukov, and A. Ankiewicz A unified approach to integrable systems via Painleve analysis by S. R. Choudhury Asymptotic stability of solitary waves for nonlinear Schrodinger equations by V. S. Buslaev and C. Sulem Finite-time blow-up in the additive supercritical stochastic nonlinear Schrodinger equations: The real noise case by A. de Bouard and A. Debussche Method of symmetry transforms for ideal magnetohydrodynamics equilibrium equations by O. I. Bogoyavlenskij The $p$-system I: The Riemann problem by R. Young Statistical analysis of collision-induced timing shifts in a wavelength-division-multiplexed optical soliton-transmission system by G. J. Morrow and S. Chakravarty Cuspons and peakons vis-a-vis regular solitons and collapse in a three-wave system by R. Grimshaw, G. A. Gottwald, and B. A. Malomed First integrals and gradient flow for a generalized Darboux-Halphen system by S. Chakravarty and R. G. Halburd Blow-ups of the Toda lattices and their intersections with the Bruhat cells by L. Casian and Y. Kodama Superposition principle for oscillatory solutions of integrable systems by M. Kovalyov Scattering at truncated solitons and inverse scattering on the semiline by H. Steudel.