A Generalized Approach To Darboux Transformations For Differential Equations

Abstract

A Darboux transformation is a mathematical procedure to produce a solution to a differential equation when the solution to a related differential equation is known. The basic idea behind a Darboux transformation is to change the discrete spectrum of a linear differential operator in a controlled way without changing its continuous spectrum. For example, by using a Darboux transformation one can describe the change in a quantum mechanical system when some of its quantum levels are removed or some extra quantum levels are added. Darboux transformation formulas for various differential equations have been developed, but such formulas seem to be specific to those particular equations without much connection among them. In our method, we develop a generalized and unified approach for Darboux transformations that is applicable to a large class of differential equations. This approach uses the solution to a linear integral equation where the kernel and nonhomogeneous terms coincide. We apply our unified approach to some specific differential equations such as the Schrodinger equation and the Zakharov-Shabat system, and we relate our results to the existing results in the literature. We also apply our results to obtain exact solutions to various integrable nonlinear partial differential equations such as the Korteweg-de Vries equation and the nonlinear Schrodinger equation.