A Method For Exact Solutions To Integrable Evolution Equations In 2+1 Dimensions

Abstract

A systematic method is developed to obtain solution formulas for certain explicit solutions to integrable nonlinear partial differential equations in two spatial variables and one time variable. The method utilizes an underlying Marchenko integral equation that arises in the corresponding inverse scattering problem. The method is demonstrated for the Kadomtsev-Petviashvili and the generalized Davey-Stewartson II system. A derivation and analysis of the solution formulas to these two nonlinear partial differential equations are given, and an independent verification of the solution formulas is presented. Such solution formulas are expressed in a compact form in terms of matrix exponentials, by using a set of four constant matrices as input. The formulas hold for any sizes of the matrix quadruplets and hence yield a large class of explicit solutions. The method presented is a generalization of a method used to find exact solutions for integrable evolution equations in one spatial variable and one time variable, which uses a constant matrix triplet as input.