On the AB system with a self-consistent source

The inverse scattering problem method is presented for solving the Cauchy problem for the AB system with a self-consistent source. The specific feature of the considered Cauchy problem is that the initial function is assumed to approach nonzero limits as the spatial variable approaches infinity. By introducing a suitable uniformization variable, the necessary information on the theory of direct and inverse scattering problems is given for a system of first-order differential equations on the two-sheeted Riemann surface of the complex plane. The analyticity, symmetry and asymptotic behavior of the Jost solutions and the scattering matrix, as well as the properties of the discrete spectrum of the scattering problem are studied. The time dependence of the scattering data was found. In particular, a one-soliton solution is obtained in the reflectionless case.