Integrability of the vector nonlinear Schrödinger–Maxwell–Bloch equation and the Cauchy matrix approach

We investigate the integrability and soliton solutions of the vector nonlinear Schrödinger–Maxwell–Bloch (VNLS–MB) equation. This equation is derived using the generalized $\bar \partial$-dressing method in a local $4\times 4$ matrix $\bar \partial$-problem. The vector nonlinear Schrödinger equation with self-consistent sources (VNLSSCS) is obtained and is proved to be equivalent to the VNLS–MB equation. Starting with Sylvester equation and the equivalence between the VNLS–MB and VNLSSCS equations, the $N$-soliton solutions of the VNLS–MB equation are successfully obtained by the Cauchy matrix approach. As an application, some interesting patterns of dynamical behavior are displayed.