Scattering Theory for Linear and Nonlinear Waves and Soliton Dynamics

线性和非线性波的散射理论和孤子动力学.Avraham Soffer摘要：分析非线性演化方程是这项工作的目标。描述波传播的非线性演化方程在许多科学和工程领域都具有重要意义。非线性薛定谔方程在多体量子系统、非线性光学等研究中自然而然地出现。在数学上，人们感兴趣的是在给定的函数类中找到所有初始数据的解的大时间行为，通常是Soblev空间。研究者和合作者以前对具有多个散射通道的演化方程所做的工作已经导致了主要的新工具，这些工具现在被应用于不同类型的方程。在适当的关于允许的非线性的假设下，现在可以描述关于非线性薛定谔方程的大时间行为的一般猜想。人们期望，对于标准Sobolev空间中的所有初始数据，渐近行为将由独立运动的孤子和自由波的组合给出。虽然这一结果超出了我们目前的能力，但在过去的几年里，研究人员及其合作者和其他人已经取得了实质性的进展，最终证明了上述大范围分离孤子的小扰动猜想。计划开发新的技术来首次处理孤子状态的大扰动。这项工作将利用许多不同的数学领域，包括调和分析、散射理论的相空间方法、非线性分析等。这一方向的进展有望对我们理解波相互作用的最重要的非线性动力系统之一起到重要作用。它适用于固体物理中的玻色爱因斯坦凝聚体、光纤中的光孤子和其他光学设备，以及研究量子场论的非微扰解等。它还将启发并将启发在色散波动方程的数学分析及其与调和和谱分析的关系方面的更多新的研究方向。

Scattering Theory for Linear and Nonlinear Waves and Soliton Dynamics.Avraham SofferAbstract: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.