On a Riemann–Hilbert problem for the NLS-MB equations

Abstract

In this paper, we study a coupled system of the nonlinear Schrödinger (NLS) equation and the Maxwell–Bloch (MB) equation with nonzero boundary conditions by Riemann–Hilbert (RH) method. We obtain the formulae of the simple-pole and the multi-pole solutions via a matrix Riemann–Hilbert problem (RHP). The explicit form of the soliton solutions for the NLS-MB equations is obtained. The soliton interaction is also given. Furthermore, we show that the multi-pole solutions can be viewed as some proper limits of the soliton solutions with simple poles, and the multi-pole solutions constitute a novel analytical viewpoint in nonlinear complex phenomena. The advantage of this way is that it avoids solving the complex symmetric relations and repeatedly solving residue conditions.