Riemann–Hilbert method for a higher-order matrix-type nonlinear Schrödinger equation with zero boundary conditions

Abstract

The objective of this research is to investigate the initial value problem of a higher-order matrix-type nonlinear Schrödinger (NLS) equation with discrete spectrum as simple and double poles (simple and second-order zeros [Formula: see text] and second- and third-order zeros under [Formula: see text] under zero boundary conditions (ZBCs). Specifically, we not only consider the analyticity, symmetries and asymptotics of the Jost function, scattering and reflection coefficients in the process of direct scattering, but also residue conditions, norming constants, RH problem and the reconstruction formula in the process of inverse scattering. There exist some differences between it and the RH method in the study of vector and scalar equations, like the order of each pole of [Formula: see text] being less than or equal to the order of the zeros of [Formula: see text] (assuming [Formula: see text] is a second- or third-order zero of [Formula: see text] under [Formula: see text]), etc.