The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method

A procedure for finding the solutions of the Vakhnenko-Parkes equation by means of the inverse scattering method is described. Both the bound state spectrum and the continuous spectrum are considered in the associated eigenvalue problem. The suggested special form of the singularity function gives rise to periodic solutions. The interaction of a soliton with a one-mode periodic wave is studied.