Energy-Dependent Reflectionless Inverse Scattering

Abstract

We develop a reflectionless inverse scattering theory regarding an energy-dependent Schrödinger equation. A necessary and sufficient condition for scattering data to recover a pair of potentials of the equation is obtained, based upon a complex generalization of a representation of reflectionless potentials via the Hirota transformation. As an application, an inverse scattering method is established for finding $N$-soliton solutions of a nonlinear evolution system that is an isospectral flow of the energy-dependent Schrödinger equation.