Affiliation:
1. Department of Mathematics , Gebze Technical University , Gebze - Kocaeli 41400 , Turkey ; and Institute of Mathematics and Mechanics, Azerbaijan National Academy of Science, 1141 Baku, Azerbaijan
Abstract
Abstract
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.