New solutions of nonstationary Schrödinger and Kadomtsev-Petviashvili equations

Summary

Abstract

A method to obtain new real, nonsingular, decaying potentials of the nonstationary Schrödinger equation and corresponding solutions of the Kadomtsev-Petviashvili equation is developed. The solutions are characterized by the order of the poles of an eigenfunction of the Schrodinger operator and an underlying topological quantity, an index or charge. The properties of some of these solutions are discussed.

Introduction

In this paper we describe a method to obtain a new class of decaying potentials and corresponding solutions to the nonstationary Schrödinger and Kadomtsev-Petviashvili-I equation (KPI; here the “I” stands for one of the two physically interesting choices of sign in the equation). These equations have significant applications in Physics. The Schrödinger operator is, of course, centrally important in quantum mechanics and the KP equation is a ubiquitous nonlinear wave equation governing weakly nonlinear long waves in two dimensions with slowly varying transverse modulation. The nonstationary Schrödinger operator can be used to linearize the KPI equation via the inverse scattering transform (IST; [see e.g. 1]). In it was shown by IST that discrete states associated with complex conjugate pairs of simple eigen-values of the Schrödinger operator yield lump type soliton solutions which decay as O(l/r²), r² = x² + y².

In it was shown, for the first time, that there are real, nonsingular, decaying potentials of the Schrödinger operator corresponding to its discrete spectrum, whose corresponding eigenfunctions have multiple poles/eigenvalues. This class of potentials is related to decaying solutions in the KPI equation which we refer to as “multipole lumps”.