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

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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.