Scattering problem for the Zakharov-Shabat equations onthe semi-axis

Abstract

This paper is concerned with the direct and inverse scatteringproblems for compatible differential equations connected withthe nonlinear Schrödinger equation (NLSE) on the semi-axis. Thecorresponding initial boundary value problem(x,t∊ℝ+) was studied recently byFokas and Its. They found thatthe key to this problem is to linearize the initial boundaryvalue problem using a Riemann-Hilbert problem. The main goal ofthis paper is to obtain characteristic properties of thescattering data for compatible differential equations. Ourapproach uses the transformation operators for both x- andt-equations. For Schwartz type initial and boundary functionswe obtain the characteristic properties (A1)-(A5) of the scatteringdata and derive the so-called xt- and t-integral equationsof Marchenko type. The xt-integral equations guarantee theexistence of the solution of the NLSE and give an expression ofthe solution with given scattering data. In turn, thet-integral equations guarantee that one can recover from thescattering data the boundary Dirichlet data v(t) and thecorresponding Neumann data w(t) consistent with the giveninitial function u(x).