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A dispersive  N -wave interaction problem (N=2⁢n{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){(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 Gelfand–Levitan–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.

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