The Cauchy problem for the (2+1) integrable nonlinear Schr\"odinger equation

Abstract

We study the Cauchy problem for the (2+1) integrable nonlinear Schr\"odinger equation by the inverse scattering transform (IST) method. This Cauchy problem with given initial data and boundary data at infinity is reduced by IST to the Cauchy problem for the linear Schr\"odinger equation, in which the potential is expressed in terms of boundary data. The results on direct and inverse scattering problems for a two-dimensional Dirac system with special potentials are used and refined. The Cauchy problem admits an explicit solution if the IST of the solution is an integral operator of rank 1. We give one such solution.