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

This paper is concerned with the direct and inverse scattering problems for compatible differential equations connected with the nonlinear Schrödinger equation (NLSE) on the semi-axis. The corresponding initial boundary value problem ( x , t ∊ + ) was studied recently by Fokas and Its. They found that the key to this problem is to linearize the initial boundary value problem using a Riemann-Hilbert problem. The main goal of this paper is to obtain characteristic properties of the scattering data for compatible differential equations. Our approach uses the transformation operators for both x - and t -equations. For Schwartz type initial and boundary functions we obtain the characteristic properties (A1)-(A5) of the scattering data and derive the so-called xt - and t -integral equations of Marchenko type. The xt -integral equations guarantee the existence of the solution of the NLSE and give an expression of the solution with given scattering data. In turn, the t -integral equations guarantee that one can recover from the scattering data the boundary Dirichlet data v ( t ) and the corresponding Neumann data w ( t ) consistent with the given initial function u ( x ).