Riemann–Hilbert approach for a Schrödinger-type equation with nonzero boundary conditions

Abstract

In this paper, we focus on investigating a nonlinear Schrödinger-type equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter [Formula: see text] into a single-valued parameter [Formula: see text]. Starting from the Lax pair of the Schrödinger-type equation, we derive its Jost solutions with nonzero boundary conditions, and further analyze the asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. An associated matrix Riemann–Hilbert (RH) problem associated with the problem of nonzero boundary conditions is subsequently presented, and a formulae of [Formula: see text]-soliton solutions for the Schrödinger-type equation by solving the matrix RH problem. As an application of the [Formula: see text]-soliton formulae, we present two kinds of one-soliton solutions and three kinds of two-soliton solutions according to different distributions of spectral parameters, and dynamical features of those solutions are also further analyzed.