N-wave soliton solution on a generic background for KPI equation

摘要

We try to generalize the inverse scattering transform (IST) forthe Kadomtsev-Petviashvili (KPI) equation to the case of potentials with“ray” type behavior, that is non-decaying along a finitenumber of directions in the plane. We present here the special butrather wide subclass of such potentials obtained by applying recursivelyN binary Backlund transformations to a decaying potential. We start witha regular rapidly decaying potential for which all elements of thedirect and inverse problem are given. We introduce an exact recursionprocedure for an arbitrary number of binary Backlund transformations andcorresponding Darboux transformations for Jost solutions and solutionsof the discrete spectrum. We show that Jost solutions obey modifiedintegral equations and present their analytical properties. We formulateconditions of reality and regularity of the potentials constructed bythese means and derive spectral data of the transformed Jost solutions.Finally we solve the recursion procedure getting a solution whichdescribes N solitons superimposed to a generic background