13 - On the Inverse Scattering Transform in Two Spatial and One Temporal Dimensions

Summary

INTRODUCTION.

The goal of this paper is to present a recently developed formalism for a) solving inverse problems in the plane for potentials decaying at infinity (i.e. given appropriate scattering data reconstruct the potential q(x,y); b) solving the initial value problem (for appropriately decaying initial data) of certain nonlinear evolution equations in two spatial and one temporal dimensions (i.e. given q(x,y,o) find q(x,y,t)). Several results obtained by this formalism are also summarized.

This formalism has been developed in a series of papers by Fokas and Ablowitz (1982a,b,c; 1983), Ablowitz et al (1982), Fokas (1982), where several inverse problems related to physically significant multidimensional equations have been formally solved. The inverse problem associated with a certain differential Riemann-Hilbert problem (in the complex x-plane), which is related to the Benjamin-Ono (BO) equation was considered in Fokas and Ablowitz (1982a). The BO equation, although an equation in 1+1 (i.e. in one space and one time dimension) has many features similar to problems in 2+1 (this results from its nonlocal character). In this sense, BO acts as a pivot from 1 + 1 to 2 + 1. The inverse problem associated with the “time“-dependent Schrb'dinger equation (see Dryuma (1974))

as well as the initial value problem of the related Kadomtsev-Petviashvili (KP)I (1970)

were considered in Fokas and Ablowitz (1982a,b). The inverse problem associated with

and the related KPII were considered in Ablowitz et al (1982). The inverse problem associated with the matrix equation

was considered by Fokas (1983) and Fokas and Ablowitz (1983). In Equation (1.5), B is a constant n x n diagonal matrix, J is a constant n x n diagonal matrix with elements either all real (hyperbolic case) or all purely imaginary (elliptic case), and q(x,y,t) is a n x n off-diagonal matrix containing the potentials (or field variables). Equation (1.5) can be used to solve several physical nonlinear equations in 2 + 1 (Ablowitz and Haberman, 19 7 5). Among them are the n-wave interaction (Ablowitz and Segur, 1981), variants of the so-called Davey-Stewartson (DS) equation (1974) (which is the long wave limi of the Benney-Eoskes equation (1969)) and the modified KP (MKP) equation.