Abstract
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.
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