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

We try to generalize the inverse scattering transform (IST) for the Kadomtsev-Petviashvili (KPI) equation to the case of potentials with “ray” type behavior, that is non-decaying along a finite number of directions in the plane. We present here the special but rather wide subclass of such potentials obtained by applying recursively N binary Backlund transformations to a decaying potential. We start with a regular rapidly decaying potential for which all elements of the direct and inverse problem are given. We introduce an exact recursion procedure for an arbitrary number of binary Backlund transformations and corresponding Darboux transformations for Jost solutions and solutions of the discrete spectrum. We show that Jost solutions obey modified integral equations and present their analytical properties. We formulate conditions of reality and regularity of the potentials constructed by these means and derive spectral data of the transformed Jost solutions. Finally we solve the recursion procedure getting a solution which describes N solitons superimposed to a generic background