On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane

Abstract

The inverse problem associated with a rather general system of n first‐order equations in the plane is linearized. When the system is hyperbolic, this is achieved by utilizing a Riemann–Hilbert problem; similarly, a ‘‘∂̄’’ (DBAR) problem is used when the system is elliptic. The above result can be employed to linearize the initial value problem associated with a variety of physically significant equations in 2+1, i.e., two spatial and one temporal dimensions. Concrete results are given for the n‐wave interaction in 2+1 and for various forms of the Davey–Stewartson equations. Lump solutions (solitons in 2+1) of the latter equation are given a definitive spectral characterization and are obtained through a linear system of algebraic equations.