On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition

On the positive semi-infinite interval, we obtained a generalization of the Marchenko method for a Dirac equation system with a discontinuous coefficient and a quadratic polynomial on a spectral parameter in the boundary condition. In this connection, we use an new integral representation of the Jost solution of equation systems, which does not have a triangular form. The scattering function of the problem is defined, and its properties are examined. The Marchenko-type main equation is obtained, and it is shown that the potential is uniquely recovered in terms the scattering function. Copyright (c) 2012 John Wiley & Sons, Ltd.