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

Abstract

A dispersive N-wave interaction problem (N = 2n), involving n velocities in two spatial and one temporal dimensions, is introduced. Explicit solutions of the problem are provided by using the inverse scattering method. The model we propose is a generalization of both the N-wave interaction problem and the (2 + 1) matrix Davey-Stewartson equation. The latter examines the Benney-type model of interactions between short and long waves. Referring to the two-dimensional Manakov system, an associated GelfandLevitan-Marchenko-type, or so-called inversion-like, equation is constructed. It is shown that the presence of the degenerate kernel reads explicit soliton-like solutions of the dispersive N-wave interaction problem. We also present a discussion on the uniqueness of the solution of the Cauchy problem on an arbitrary time interval for small initial data.