ABSTRACT
The Volterra integral equations for the unknown boundary data are derived. The solutions of these derived equations are found by the method of successive approximations in terms of known initial and boundary conditions of the considered initial-boundary value problems (IBVP). Then the solutions of the Dirichlet IBVPs are found and expressed through the solution of the system of fundamental equations in the inverse problem. The known functions in this system are constructed from the scattering data. The sine-Gordon equation in light-cone coordinates is one of the most widely studied nonlinear wave equations, because of its intrinsic mathematical properties and its wide applicability in physics. The scattering problem with the non-scattering potential is completely determined by the characteristics of the discrete spectrum.
