About the Camassa–Holm equation with a self-consistent source

This work is devoted to solving the Camassa–Holm equation with a self-consistent source of a special type by the inverse scattering method. The main result consists in determining the evolution of the scattering data for the spectral problem associated with the Camassa–Holm equation with a self-consistent source of a special type. In contrast to the classical Camassa–Holm equation, the eigenvalues of the spectral problem are moving in problem under consideration. The resulting equalities determine the evolution of the scattering data completely; this fact allows us to apply the inverse scattering method for solving the considered problem.