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

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640395/
• 关键词: Davey-Stewartson Equation; Darboux transformation; Homoclinic solution; Plane wave solution; Growth of disturbance; Nonlinear satulation; Linear stability analysis; 非線形飽和解; 平面波解の攪乱の時間成長; 高次元可積分方程式; 安定性; ラックス方程式

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.

关于多维中非线性现象稳定性的分析尚未得到充分研究，尽管将非线性整合系统的理论应用于更高的维度很重要。 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.其次，我们研究了平面波造成的干扰的时间发展。结果，我们得出了扰动的增长率的关系。我们发现，在通常的边界条件下，在时间的过程中可以忽略边界上的小波动，而干扰的生长仅取决于平面波溶液的波数和干扰。我们得出了DARBOUX-type Type Transformation的DS方程，以一种有用的形式，可用于得出同源物质溶液。为了避免LAX对依赖对空间衍生物的依赖性的复杂性，我们为JOST函数引入了其他条件，这反映了DS与非线性Schrodinger方程之间的关系以及DS方程的结构。最后，从平面波解决方案和darboux型变换中得出了一些新型溶液的明确表达。