Special singularity function for continuous part of the spectral data in the associated eigenvalue problem for nonlinear equations

The procedure for finding the solutions of the Vakhnenko-Parkes equation by means
of the inverse scattering method is described. The continuous spectrum is taken into
account in the associated eigenvalue problem. The suggested special form of the
singularity function for continuous part of the spectral data gives rise to the multimode solutions. The sufficient conditions are proved in order that these solutions become real functions. The interaction of the N periodic waves is studied. The procedure is illustrated by considering a number of examples