Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

doi:10.1007/s42967-024-00418-6

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 177-194.doi: 10.1007/s42967-024-00418-6

Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

Jing Shen¹, Huan Liu²

1. School of Sciences, Henan University of Technology, Zhengzhou, 450001, Henan, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, Henan, China

收稿日期:2024-03-20 修回日期:2024-04-26 出版日期:2026-02-20 发布日期:2026-02-11
通讯作者: Huan Liu,E-mail:liuhuan@zzu.edu.cn E-mail:liuhuan@zzu.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12171439, 12101190).

Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

Jing Shen¹, Huan Liu²

1. School of Sciences, Henan University of Technology, Zhengzhou, 450001, Henan, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, Henan, China

Received:2024-03-20 Revised:2024-04-26 Online:2026-02-20 Published:2026-02-11
Contact: Huan Liu,E-mail:liuhuan@zzu.edu.cn E-mail:liuhuan@zzu.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12171439, 12101190).

摘要/Abstract

摘要： We investigate the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with a particular class of nonvanishing boundary conditions (NVBCs), especially in the case of reflectionless potentials that give rise to a transmission coefficient with an arbitrary finite number N pairs of higher-order poles. The inverse problem is characterized in terms of a 2×2 matrix Riemann-Hilbert (RH) problem equipped with several residue conditions at N pairs of higher-order poles. In the reflectionless case, we point out that the N-multipole soliton solutions including higher-order Kuznetsov-Ma breathers and Akhmediev breathers can be reconstructed by a linear algebraic system. Furthermore, we verify these special solutions by numerical simulations and display their density structures, also derive some Peregrine solitons by choosing appropriate parameters and taking limits.

关键词: Nonlinear Schrödinger (NLS) equation, Nonvanishing boundary conditions (NVBCs), Riemann-Hilbert (RH) problem, N-multipole soliton solutions

Abstract: We investigate the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with a particular class of nonvanishing boundary conditions (NVBCs), especially in the case of reflectionless potentials that give rise to a transmission coefficient with an arbitrary finite number N pairs of higher-order poles. The inverse problem is characterized in terms of a 2×2 matrix Riemann-Hilbert (RH) problem equipped with several residue conditions at N pairs of higher-order poles. In the reflectionless case, we point out that the N-multipole soliton solutions including higher-order Kuznetsov-Ma breathers and Akhmediev breathers can be reconstructed by a linear algebraic system. Furthermore, we verify these special solutions by numerical simulations and display their density structures, also derive some Peregrine solitons by choosing appropriate parameters and taking limits.

Key words: Nonlinear Schrödinger (NLS) equation, Nonvanishing boundary conditions (NVBCs), Riemann-Hilbert (RH) problem, N-multipole soliton solutions

中图分类号:

37K10

Jing Shen, Huan Liu. Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 177-194.

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Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

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