Stability and Long-Time Behavior of Hamiltonian Partial Differential Equations

这个项目研究数学物理中的哈密顿偏微分方程解的稳定性和长时间行为，它被看作是无限维空间上的动力系统。该项目专注于两种不同的方法来研究孤立波的稳定性。第一种方法是将演化形式的方程线性化，并基于波速的某一函数的凸性来证明稳定性。第二种方法是用变分方法同时证明孤立波的存在性和稳定性。该项目着重研究与衍射管理波导阵研究相关的非线性薛定谔方程、色散管理非线性薛定谔方程和离散非线性薛定谔方程的稳定性。该项目还将研究Benjamin-Bona-Mahone型和Camassa-Holm型方程的吸引子的存在性和正则性。大多数重要的物理现象都由偏微分方程组描述，其中一些可以被视为无限维空间上的动力系统。这项工作的目的是通过这个观点来分析在许多物理情况下出现的一类偏微分方程解的定性行为。本文研究的一个主要例子是描述电磁波在光波导中传播的包络方程。作为非线性和色散效应之间平衡的结果，该方程具有被称为光孤子的脉冲式解。光孤子的研究是非线性科学中最令人兴奋的研究领域之一，因为自导波(孤子)是用于下一代光通信网络的理想候选。这项研究将加深我们对这一现象和其他具有重要物理意义的非线性模型中的波和相干结构的理解。该项目还将使本科生和研究生能够参与现代跨学科数学。

This project studies the stability and long time behavior of distinguished solutions to Hamiltonian partial differential equations of mathematical physics, viewed as dynamical systems on an infinite dimensional space. The project focuses on two different methods for investigating the stability of solitary waves. The first technique is to use linearization of the equation written in evolution form and to prove stability based on convexity of a certain function of the wave speed. The second technique is to use variational methods to simultaneously establish both existence and stability of solitary waves. The project emphasizes study of stability in the nonlinear Schrodinger equations, dispersion managed nonlinear Schrodinger equations, and discrete nonlinear Schrodinger equations that are related to the study of diffraction managed waveguide arrays. The project will also investigate existence and regularity of attractors for Benjamin-Bona-Mahoney and Camassa-Holm type equations.Most important physical phenomena are described by partial differential equations, some of which can be viewed as dynamical systems on an infinite dimensional space. The goal of this work is to analyze, via this viewpoint, the qualitative behavior of solutions to a class of partial differential equations that arise in numerous physical situations. A primary example under study in this work is the envelope equation describing an electromagnetic wave propagating in an optical waveguide. As a consequence of a balance between nonlinear and dispersive effects, the equation possesses pulse-like solutions called optical solitons. Research in the field of optical solitons is one of the most exciting areas of research in nonlinear science, since self-guided waves (solitons) are ideal candidates for use in the next generation of optical communication networks. This research will deepen our understanding of the waves and coherent structures in nonlinear models of this phenomenon and others of physical importance. The project will also enable undergraduate and graduate students to be involved in modern interdisciplinary mathematics.