tailieunhanh - Scattering data in an inverse scattering problem on the semi-axis for a first-order hyperbolic system

The inverse scattering problem for the first-order hyperbolic system on the semi-axis in the case of 2 incident and 2 scattered waves under consideration of 2 problems with the same given incident waves and different boundary conditions is considered. The scattering data on the semi-axis are given in terms of the scattering operator on the whole axis for the same system with the coefficients, which are extended in the whole axis by zero. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 110 – 118 ¨ ITAK ˙ c TUB ⃝ doi: Scattering data in an inverse scattering problem on the semi-axis for a first-order hyperbolic system ˙ ˙ Mansur ISGENDEROGLU ISMAILOV∗, Ibrahim TEKIN Department of Mathematics, Gebze Institute of Technology, Gebze, Kocaeli, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: The inverse scattering problem for the first-order hyperbolic system on the semi-axis in the case of 2 incident and 2 scattered waves under consideration of 2 problems with the same given incident waves and different boundary conditions is considered. The scattering data on the semi-axis are given in terms of the scattering operator on the whole axis for the same system with the coefficients, which are extended in the whole axis by zero. Key words: Inverse scattering problem, scattering data, first-order hyperbolic system 1. Introduction The inverse scattering problem (ISP) for differential equations is the problem of finding their unknown coefficients from the scattering operator or scattering data. Such an ISP must satisfy the following requirements: the solution must be unique, the algorithm of the recovering of the coefficients must be given, and the characterization of the scattering data must be determined. The scattering data are the minimal information from which the ISP can be uniquely solved. Since the number of known functions in the ISP is greater than the number of unknown coefficients of differential equations, selection of the minimal information is important for multidimensional inverse problems. The ISP on the whole axis for the first-order hyperbolic system was solved in [4] via the Gelfand– Levitan–Marchenko (GLM) equation and the scattering data for the ISP were given in [7,9]. The nonlocal Riemann–Hilbert (RH) approach to the ISP for the .