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

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Abstract
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).