Exact solutions of the Korteweg-de Vries equation with a self-consistent source

Exact solutions are found for the Korteweg-de Vries equation with a source satisfying the stationary Schroedinger equation. Each solution describes the evolution of the initial moving wave with one phase velocity to the final wave with another phase velocity. The conditions are pointed out under which the phase velocities of these waves may differ in sign. The obtained results are relevant to some problems of hydrodynamics, plasma physics, solid state physics, etc