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

将非线性可积系统的理论应用于高维是很重要的，但对多维非线性现象的稳定性分析研究还不够充分。本研究项目的目的是通过推导二维典型可积模型之一Davey-Stewartson （DS）方程的同斜型解，为这种分析奠定基础。为了推导同斜解，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，我们研究了平面波扰动的时间发展。由此，我们推导出了扰动增长率的关系式。我们发现，在通常的边界条件下，边界上的小波动在时间过程中可以忽略不计，扰动的增长仅由平面波解的波数和扰动决定。其次，我们导出了DS方程的darboux型变换，该变换的形式有助于导出同斜解。为了避免Lax对空间导数算子依赖的复杂性，我们引入了Jost函数的附加条件，反映了DS与非线性薛定谔方程的关系，以及DS方程的结构。最后，从平面波解和darboux型变换中导出了一些新型解的显式表达式。

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.