Stability and Long-Time Behavior of Hamiltonian Partial Differential Equations

哈密​​顿偏微分方程的稳定性和长期行为

• 批准号: 0508184
• 负责人: Milena Stanislavova
• 金额: $ 11.62万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0508184&HistoricalAwards=false
• 关键词: Stability; Long; Time; Behavior; Hamiltonian

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.

本计画研究数学物理中的哈密尔顿偏微分方程，视其为无限维空间上的动力系统，其特徴解的稳定性与长时间行为。 该项目侧重于两种不同的方法来研究孤立波的稳定性。 第一种技术是使用线性化的方程写在演化形式，并证明稳定性的基础上的一个特定的函数的波速凸。 第二种方法是利用变分方法同时建立孤立波的存在性和稳定性。 本项目的重点是研究与衍射管理波导阵列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-Mahoney和Camassa-Holm型方程吸引子的存在性和规律性。大多数重要的物理现象都由偏微分方程描述，其中一些可以被视为无限维空间上的动力系统。 这项工作的目标是分析，通过这种观点，定性行为的解决方案，一类偏微分方程，出现在许多物理情况。 在这项工作中研究的一个主要例子是描述在光波导中传播的电磁波的包络方程。 由于非线性效应和色散效应之间的平衡，该方程具有称为光孤子的脉冲状解。 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）是下一代光通信网络的理想候选者。 这项研究将加深我们对这种现象和其他物理重要性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参与现代跨学科数学。