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N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
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Symmetry 2019, 11(6), 826; https://doi.org/10.3390/sym11060826
Received: 6 June 2019 / Revised: 17 June 2019 / Accepted: 20 June 2019 / Published: 22 June 2019
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Abstract

In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of det P + and det P ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R–H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R–H problem, the evolution equation of the R–H data with the perturbation is derived. View Full-Text
Keywords: NLS-like equation; Riemann–Hilbert problem; symmetry; N-soliton solutions; R–H data NLS-like equation; Riemann–Hilbert problem; symmetry; N-soliton solutions; R–H data
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Lin, Y.; Dong, H.; Fang, Y. N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem. Symmetry 2019, 11, 826.

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