Inverse scattering method for an integrable system of derivative nonlinear Schr\"odinger equations

TL;DR

This paper develops an inverse scattering method for solving the derivative nonlinear Schrödinger II system, enabling the recovery of potentials and solutions through a Marchenko integral equation approach.

Contribution

It introduces a novel inverse scattering technique using a Marchenko system for the DNLS II system, incorporating matrix triplet data for potential reconstruction.

Findings

Successfully solves the inverse scattering problem for the DNLS II system.

Recovers time-evolved potentials from input data.

Provides a systematic method for solving the integrable system.

Abstract

We present a solution method for the integrable system (derivative nonlinear Schr\"odinger II system) or the Chen--Lee--Liu system. This is done by presenting a solution technique for the inverse scattering problem for the corresponding linear system of ordinary differential equations with energy-dependent potentials. The relevant inverse scattering problem is solved by establishing a system of linear integral equations, which we refer to as the Marchenko system of linear integral equations. In solving the inverse scattering problem we use the input data set consisting of a transmission coefficient, a reflection coefficient, and the bound-state information presented in the form of a pair of matrix triplets. Using our data set as input to the Marchenko system, we recover the potentials from the solution to the Marchenko system. By using the time-evolved input data set, we recover the time-evolved potentials, where those potentials form a solution to the integrable DNLS II system.