Project/Area Number 
13640395

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
物理学一般

Research Institution  Utsunomiya University 
Principal Investigator 
YAJIMA Tetsu Utsunomiya University, Department Information Science, 工学部, 助教授 (40230198)

CoInvestigator(Kenkyūbuntansha) 
UJINO Hideaki Gunma College of Technology, Associate Professor, 助教授 (00321399)
NISHINARI Katsuhiro Ryukoku University, Department of Applied Mathematics and Informatics, Associate Professor, 理工学部, 助教授 (40272083)

Project Period (FY) 
2001 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2003: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)

Keywords  DaveyStewartson 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 
Research Abstract 
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 DaveyStewartson (DS) equation, which is one of the typical integrable models in twodimensions. 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 Darbouxtype 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 Darbouxtype transform have derived.
