Scattering Theory for Linear and Nonlinear Waves and Soliton Dynamics

Scattering Theory for Linear and Nonlinear Waves and Soliton Dynamics. Avraham Soffer Abstract: The analysis of nonlinear evolution equations is the goal of this work. Nonlinear evolution equations which describe wave propagation are of fundamental importance in many fields of science and engineering. The nonlinear Schroedinger equation appears naturally in the study of many body quantum system, nonlinear optics and more. Mathematically, one is interested in finding the large time behavior of solutions for all initial data in a given class of functions, typically a Sobolev space. The previous works on evolution equations which have many channels of scattering , by the Investigator and collaborators, have led to major new tools which are now applied to different types of equations. Under suitable assumptions on the class of allowed nonlinearities one can now state the general conjecture about the large time behavior of the nonlinear Schroedinger equation. One expects that for all initial data in the standard Sobolev space, the asymptotic behavior will be given by a combination of independently moving solitons and a free wave. While this result is beyond our current capabilities, substantial progress has been made in the last few years, by the Investigator his collaborators and others, culminating in the proof of the above conjecture for small perturbations of widely separated solitons. It is planned to develop new techniques to deal for the first time with large perturbations of soliton states. This effort will draw on many and diverse fields of mathematics, including harmonic analysis, phase space methods of scattering theory, nonlinear analysis and more. The advances in this direction are expected to play an important role in our understanding of one of the most important nonlinear dynamical systems of wave interactions. It applies to Bose Einstein condensates in solid state physics, optical solitons in fibers and other optical devices, in the study of nonperturbative solutions to Quantum Field Theory and more. It also inspires and will inspire more new directions of research in the mathematical analysis of dispersive wave equations, and its relation to harmonic and spectral analysis.