The Marchenko method to solve the general system of derivative nonlinear Schr\"odinger equations

TL;DR

This paper develops a Marchenko integral equation approach to solve the inverse scattering problem for the derivative nonlinear Schrödinger equations, providing explicit formulas and analysis for potentials and solutions.

Contribution

It introduces a new integral equation system for the inverse scattering problem of derivative NLS equations, including explicit solution formulas and analysis of special cases.

Findings

Explicit solutions for potentials and Jost functions when reflection coefficients are zero.

Derivation of reduced Marchenko equations for related potentials.

Application of the method to obtain solutions for the derivative NLS equation.

Abstract

A system of linear integral equations is presented, which is the analog of the system of Marchenko integral equations, to solve the inverse scattering problem for the linear system associated with the derivative NLS equations. The corresponding direct and inverse scattering problems are analyzed, and the recovery of the potentials and the Jost solutions from the solution to the Marchenko system is described. When the reflection coefficients are zero, some explicit solution formulas are provided for the potentials and the Jost solutions in terms of a pair of constant matrix triplets representing the bound-state information for any number of bound states and any multiplicities. In the reduced case, when the two potentials in the linear system are related to each other through complex conjugation, the corresponding reduced Marchenko integral equation is obtained. The solution to the derivative NLS equation is obtained from the solution to the reduced Marchenko integral equation. The theory presented is illustrated with some explicit examples.