Integration of the negative order Korteweg-de Vries equation with a special source

In this paper, we consider the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem. It is shown that the considered equation can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm-Liouville operator with a periodic potential associated with the solution of the considered equation is determined. The results obtained make it possible to apply the inverse problem method for solving the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem.