The inverse problem for some special spectral data

Abstract

In this paper the spectral problem of third order for the inverse scattering transform (IST) method is solved. For the discrete part of the spectral data, the two-multiple poles are taken into account. The line spectrum of continuum states for the IST method is examined as well. The suggested spectrum approximates in first order the step-function. The scope for the suggested spectral data is demonstrated through the analysis of the Vakhnenko–Parkes equation that allows new solutions to be obtained. The account of the time-dependence is different from the standard procedure.