Matrix Zakharov-Shabat Systems with Zero Diagonal Entry

TL;DR

This paper develops the scattering theory for a specific AKNS system with a zero diagonal entry, enabling solutions to certain long-wave-short-wave equations via inverse scattering.

Contribution

It introduces a novel scattering framework for the AKNS system with a zero diagonal entry, expanding the inverse scattering method to this new class.

Findings

Derived the direct and inverse scattering transforms for the system.

Established the time evolution of scattering data.

Solved the initial-value problem for related long-wave-short-wave equations.

Abstract

In this article we develop the direct and inverse scattering theory of the Ablowitz-Kaup-Newell-Segur (AKNS) system $\bv_x=(ik\zS+\CQ(x))\bv$, where $\zS$ is a diagonal $n\times n$ matrix with diagonal entries $1$ and $-1$ and a single zero diagonal entry and $\CQ(x)$ is an $n\times n$ potential anticommuting with $\zS$ with entries in $L^1(\R)$. We derive the time evolution of the scattering data which, through the inverse scattering transform, lead to the solution of the initial-value problem for a system of long-wave-short-wave equations.