Matrix Zakharov-Shabat Systems with Zero Diagonal Entry

TL;DR

The paper develops a direct and inverse scattering framework for the matrix AKNS system with a zero diagonal entry, specified by $\Sigma=\operatorname{diag}(I_{m_+},0,-I_{m_-})$ and $\mathcal{Q}(x)\in L^1(\mathbb{R})$ anticommuting with $\Sigma$, extending existing results to arbitrary $n\ge3$. It resolves the incompleteness of the Jost-solution set by constructing two missing Jost solutions via dual systems and wedge products, and presents both Marchenko and Riemann–Hilbert formulations of the inverse problem, together with a Wiener-algebra treatment of the involved data. The inverse scattering transform then yields the solution to the initial-value problem for a generalized long-wave-short-wave system, with explicit time evolution of the scattering data. Time dependence is derived from the Lax pair, showing that corrected transmission data are time-invariant while reflection data and norming constants evolve through explicit exponentials in $t$, enabling full IST-based reconstruction of the potential for all times.

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.