Inverse scattering transform for N-wave interaction problem with a dispersive term in two spatial dimensions

TL;DR

This paper introduces a dispersive N-wave interaction problem in two spatial dimensions, providing exact solutions and a novel inverse scattering method, extending classical models to higher dimensions with soliton solutions.

Contribution

It generalizes the N-wave interaction problem and matrix Davey-Stewartson equation to 2+1 dimensions, constructing an inverse scattering framework and soliton solutions for the dispersive case.

Findings

Exact solutions of the dispersive N-wave interaction problem are obtained.

A Gelfand-Levitan-Marchenko-type equation is constructed for the 2D analog of the Manakov system.

Unique solutions for small initial data are established for arbitrary time intervals.

Abstract

In this work, we introduce a dispersive N(=2n)-wave interaction problem involving n velocities in two spatial dimensions and one temporal dimension. Exact solutions of the problem are exhibited. This is a generalization of the N-wave interaction problem and matrix Davey-Stewartson equation with 2+1 dimensions that examines the Benney-type model of interactions between short and long waves. Accordingly, associated with the solutions of two dimensional analog of the Manakov system, a Gelfand-Levitan-Marchenko (GLM)-type, or so-called inversion-like, equation is constructed. It is shown that the presence of the degenerate kernel reads exact soliton-like solutions of the dispersive N-wave interaction problem.We also mention the unique solution of the Cauchy problem on an arbitrary time interval for small initial data.