Localized And Homoclinic Solutions of a Nonlinear Wave Equation in Two-Dimensional Space

二维空间中非线性波动方程的定域同宿解

基本信息

批准号：
13640395
负责人：
YAJIMA Tetsu
金额：
$ 1.66万
依托单位：
Utsunomiya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

The analysis on stability of nonlinear phenomena in multidimensions has not been studied sufficiently, although it is important to apply the theory of nonlinear integrable system to higher dimensions. The purpose of this research project is to establish a basis of such an analysis by deriving homoclinic type solutions for the Davey-Stewartson (DS) equation, which is one of the typical integrable models in two-dimensions.In order to derive homoclinic solutions, we analyzed the plane wave solution and associated Jost functions for the DS equation, and found that the growth rate of the Jost functions has given in terms of the wave number. Secondly, we have studied the time development of the disturbance caused in the plane wave. As a result, we derived a relation of the growth rate of the perturbation. We have found that the small fluctuations on the boundary can be neglected in the course of time under usual boundary conditions, and the growth of disturbance is determined only by the wave number of the plane wave solutions and the disturbance.Next, we derived the Darboux-type transform for the DS equations in a form which is useful to derive homoclinic solutions. To avoid the complexity of the dependence of Lax pairs on space derivative operators, we have introduced an additional conditions for Jost functions, which reflects the relation between the DS and the nonlinear Schrodinger equation, and the structures of the DS equation. Finally, some explicit expressions of new types of solutions from the plane wave solutions and Darboux-type transform have derived.

尽管将非线性可积系统理论应用于更高维度很重要，但多维非线性现象的稳定性分析尚未得到充分研究。本研究项目的目的是通过推导 Davey-Stewartson (DS) 方程的同宿型解来建立此类分析的基础，该方程是典型的二维可积模型之一。为了推导同宿解，我们分析了 DS 方程的平面波解和相关的 Jost 函数，发现 Jost 函数的增长率以波数的形式给出。其次，我们研究了平面波引起的扰动的时间发展。结果，我们得出了扰动增长率的关系。我们发现，在通常的边界条件下，随着时间的推移，边界上的微小波动可以忽略不计，扰动的增长仅由平面波解的波数和扰动决定。接下来，我们推导了达布DS 方程的类型变换，其形式有助于导出同宿解。为了避免Lax对对空间导数算子依赖的复杂性，我们为Jost函数引入了一个附加条件，它反映了DS与非线性薛定谔方程之间的关系，以及DS方程的结构。最后，推导了平面波解和达布型变换的新型解的一些显式表达式。