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.

中文翻译：

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关于 NLS-MB 方程的 Riemann-Hilbert 问题

在本文中，我们通过黎曼-希尔伯特（RH）方法研究了具有非零边界条件的非线性薛定谔（NLS）方程和麦克斯韦-布洛赫（MB）方程的耦合系统。我们通过矩阵 Riemann-Hilbert 问题 (RHP) 获得了单极点和多极点解的公式。得到了 NLS-MB 方程的孤子解的显式形式。还给出了孤子相互作用。此外，我们证明了多极点解可以看作是具有简单极点的孤子解的一些适当极限，并且多极点解构成了非线性复杂现象的新分析观点。这种方式的优点是避免了求解复杂的对称关系和重复求解剩余条件。