Scattering and Inverse Scattering on the Line for a First-Order System with Energy-Dependent Potentials

Abstract

A first-order system of two linear ordinary differential equations is analyzed. The linear system contains a spectral parameter, and it has two coefficients that are functions of the spatial variable 𝑥. Those two functions act as potentials in the linear system and they also linearly contain the spectral parameter λ, and hence they are referred to as energy-dependent potentials. Such a linear system arises in the solution to a pair of integrable nonlinear partial differential equations (known as the derivative nonlinear Schrödinger equations) via the so-called inverse scattering transform method.

The direct and inverse problems for the corresponding first-order linear system with energy-dependent potentials are investigated. In the direct problem, when the two potentials belong to the Schwartz class, the properties of the corresponding scattering coefficients and so-called bound-state data are derived. In the inverse problem, the two potentials are recovered from the scattering data set consisting of the scattering coefficients and bound-state data. The solutions to the direct and inverse problems are achieved by relating the scattering data and the potentials in the energy-dependent system to those in a pair of first-order system with energy-independent potentials. An alternate solution to the inverse problem is given by formulating a linear integral equation (referred to as the alternate Marchenko integral equation), and the energy-dependent potentials are recovered with the help of the solution to the alternate Marchenko equation.