This paper is concerned with the modified nonlinear Schrödinger equation with zero boundary condition or non-zero boundary condition, which models Alfvén wave propagating along a magnetic field in cold plasmas. We are interested in the double-pole solutions that are a kind of bound-state solitons. By solving a direct scattering problem, the Jost eigenfunctions and scattering matrix are given, their symmetries and asymptotic behaviors are also presented. Then the inverse scattering problems are solved in terms of the matrix Riemann–Hilbert method. In addition, the trace formulae for analytic scattering coefficients and theta conditions are derived. Finally, the explicit formulae of double-pole solutions for the equation are obtained.

本文关注具有零边界条件或非零边界条件的修正非线性薛定谔方程，它模拟了在冷等离子体中沿磁场传播的阿尔文波。我们感兴趣的是一种束缚态孤子的双极解。通过求解直接散射问题，给出了Jost特征函数和散射矩阵，并给出了它们的对称性和渐近行为。然后根据矩阵 Riemann–Hilbert 方法求解逆散射问题。此外，推导了解析散射系数和θ条件的迹线公式。最后得到方程双极解的显式。