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

虽然将非线性可积系统理论应用于高维系统具有重要意义，但对高维系统中非线性现象的稳定性分析还不够深入。本文的研究目的是通过推导二维可积模型之一Davey-Stewartson（DS）方程的同宿型解，为这一分析奠定基础。为了得到同宿型解，我们分析了DS方程的平面波解和相应的Jost函数，并发现Jost函数的增长率已用波数给出。其次，我们研究了平面波中扰动的时间发展。由此导出了微扰增长率的关系式。我们发现，在通常的边界条件下，边界上的小扰动随时间的推移可以忽略，扰动的增长仅由平面波解的波数和扰动决定.其次，我们导出了DS方程的Darboux型变换的形式，它对导出同宿解是有用的.为了避免Lax对依赖于空间导数算子的复杂性，我们引入了Jost函数的一个附加条件，它反映了DS方程与非线性Schrodinger方程的关系以及DS方程的结构.最后，从平面波解和Darboux型变换出发，导出了一些新的解的显式表达式。

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.