Matrix Zakharov-Shabat Systems with Zero Diagonal Entry

In this article we develop the direct and inverse scattering theory of the Ablowitz-Kaup-Newell-Segur (AKNS) system ${\text{bv}}_x=(ik\text{zS}+\text{CQ}(x))\text{bv}$, where $\text{zS}$ is a diagonal $n\times n$ matrix with diagonal entries $1$ and $-1$ and a single zero diagonal entry and $\text{CQ}(x)$ is an $n\times n$ potential anticommuting with $\text{zS}$ with entries in $L^1(\mathbb{R})$. 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.