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

School of Mathematical Sciences, Institute of Mathematics and Key Laboratory, of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, P. R. China

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and

Engui Fan

https://orcid.org/0000-0002-8816-8398

School of Mathematical Sciences, Institute of Mathematics and Key Laboratory, of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, P. R. China

E-mail Address: faneg@fudan.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0217984921502080Cited by:8 (Source: Crossref)

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 $k$ $k$ into a single-valued parameter $z$ $z$ . 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 $N$ $N$ -soliton solutions for the Schrödinger-type equation by solving the matrix RH problem. As an application of the $N$ $N$ -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.

Keywords:

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