From the AKNS system to the matrix Schroedinger equation with vanishing potentials: Direct and inverse problems

TL;DR

This paper establishes a connection between the AKNS scattering theory and the matrix Schrödinger equation with vanishing potentials, introduces a Miura transformation to relate the focusing matrix NLS to a nonlocal integrable equation, and develops a new multisoliton solution method.

Contribution

It presents a novel link between AKNS and matrix Schrödinger systems, introduces a Miura transformation for nonlocal equations, and proposes a new approach to solve the matrix NLS equation.

Findings

Derived multisoliton solutions for the nonlocal integrable equation.

Established a connection between focusing AKNS system and matrix Schrödinger equation.

Proposed a new method for solving the matrix NLS equation.

Abstract

We relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schroedinger equation. The corresponding Miura transformation which allows this connection, converts the focusing matrix nonlinear Schroedinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.