具有非零边界条件的混合 Chen-Lee-Liu 导数非线性薛定谔方程的单极解和双极解

数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 101-122.

具有非零边界条件的混合 Chen-Lee-Liu 导数非线性薛定谔方程的单极解和双极解

汪春江^1,^*(),张健²()

¹四川师范大学数学科学系成都 610100
²电子科技大学数学科学系成都611731

收稿日期:2021-12-29 修回日期:2022-08-09 出版日期:2023-02-26 发布日期:2023-03-07
通讯作者: ^*汪春江, E-mail: wangchunjiangmath@163.com
作者简介:张健, E-mail: zhangjian@uestc.edu.cn
基金资助:
国家自然科学基金(11871138)

Simple-Pole and Double-Pole Solutions for the Mixed Chen-Lee-Liu Derivative Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

Wang Chunjiang^1,^*(),Zhang Jian²()

¹School of Mathematical Sciences, Sichuan Normal University, Chengdu 610100
²School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731

Received:2021-12-29 Revised:2022-08-09 Online:2023-02-26 Published:2023-03-07
Supported by:
The NSFC(11871138)

摘要/Abstract

摘要：

该文研究在无穷远处具有非零边界条件的混合 Chen-Lee-Liu 导数非线性薛定谔方程的单极解和双极解. 通过求解直散射问题, 得到了 Jost 解和散射矩阵, 并给出了它们的对称性和渐近性. 然后, 利用矩阵 Riemann-Hilbert 方法求解逆散射问题. 此外, 还得到了解析散射系数的迹公式和 $\theta$ 条件. 最后, 得到了该方程的单极解和双极解的显式表达式.

关键词: 非线性薛定谔方程, 非零边界条件, 逆散射, Riemann-Hilbert 方法, 双极解

Abstract:

This paper is concerned with simple-pole and double-pole solutions for the mixed Chen-Lee-Liu derivative nonlinear Schr?dinger equation with non-zero boundary conditions at infinity. By solving a direct scattering problem, the Jost eigenfunctions and scattering matrix are given, their symmetries and asymptotic behaviors are also presented. Then the inverse scattering problems are solved in terms of the matrix Riemann-Hilbert method. In addition, the trace formulae for analytic scattering coefficients and theta conditions are derived. Finally, the explicit formulae of double-pole solutions for the equation are obtained.

Key words: Nonlinear Schr?dinger equation, Non-zero boundary conditions, Inverse scattering, Riemann-Hilbert problem, Double-pole solution

中图分类号:

O175

汪春江, 张健. 具有非零边界条件的混合 Chen-Lee-Liu 导数非线性薛定谔方程的单极解和双极解[J]. 数学物理学报, 2023, 43(1): 101-122.

Wang Chunjiang, Zhang Jian. Simple-Pole and Double-Pole Solutions for the Mixed Chen-Lee-Liu Derivative Nonlinear Schrödinger Equation with Nonzero Boundary Conditions[J]. Acta mathematica scientia,Series A, 2023, 43(1): 101-122.

图/表 2

参考文献 17

[1]	Kundu A. Landau-Lifshitz and higherorder nonlinear systems gauge generated from nonlinear Schrödinger type equations. J Math Phys, 1984, 25(12): 3433-3438 doi: 10.1063/1.526113
[2]	Kivshar Y S, Agrawal G P. Optical Solitons:From Fibers to Photonic Crystals. New York: Academic, 2003
[3]	Dysthe K B. Note on the modification of the nonlinear Schödinger equation for application to deep water waves. Proc R Soc Lond A, 1979, 369(1736): 105-114 doi: 10.1098/rspa.1979.0154
[4]	Chan H N, Chow K W, Kedziora D J. Rogue wave modes for a derivative nonlinear Schrödinger model. Phys Rev E, 2014, 89(3): 032914 doi: 10.1103/PhysRevE.89.032914
[5]	Hu B B, Zhang L, Zhang N. On the Riemann-Hilbert problem for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation. J Comput Appl Math, 2021, 390: 113393 doi: 10.1016/j.cam.2021.113393
[6]	Zhang Y S, Guo L J, Chabchoub A. Higher-order rogue wave dynamics for a derivative nonlinear Schrödinger equation. https://arxiv.org/pdf/1409.7923v2.pdf
[7]	Fang F, Hu B B, Zhang L. Riemann-Hilbert method and $N$-soliton solutions for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation. https://arxiv.org/pdf/2004.03193.pdf
[8]	Zhao Y, Fan E G. N-soliton solution for a higher-order Chen-Lee-Liu equation with nonzero boundary conditions. Modern Phys Lett B, 2020, 34(4): 2050054 doi: 10.1142/S0217984920500542
[9]	Biondini G, Kovačič G. Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J Math Phys, 2014, 55(3): 031506 doi: 10.1063/1.4868483
[10]	Pichler M, Biondini G. On the focusing non-linear Schrödinger equation with non-zero boundary conditions and double poles. IMA J Appl Math, 2017, 82(1): 131-151 doi: 10.1093/imamat/hxw009
[11]	Wen L L, Zhang N, Fan E G. $N$-soliton solution of the Kundu-Type equation via Riemann-Hilbert approach, Acta Mathematica Scientia, 2020, 40B(1): 113-126
[12]	Zhang B, Fan E G. Riemann-Hilbert approach for a Schrödinger-type equation with nonzero boundary conditions. Modern Phys Lett B, 2021, 35(12): 2150208 doi: 10.1142/S0217984921502080
[13]	Zhang G Q, Yan Z Z. The derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions: Inverse scattering transforms and N-double-pole solutions. J Nonlinear Sci, 2020, 30(2): 3089-3127 doi: 10.1007/s00332-020-09645-6
[14]	Zhang G Q, Yan Z Z. Focusing and defocusing Hirota equations with non-zero boundary conditions: Inverse scattering transforms and soliton solutions. Commun Nonlinear Sci Numer Simulat, 2020, 80: 104927 doi: 10.1016/j.cnsns.2019.104927
[15]	Zhang G Q, Yan Z Z. Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions. Physica D, 2020, 410: 132521 doi: 10.1016/j.physd.2020.132521
[16]	Clarkson P A, Cosgrove C M. Painlevé analysis of the non-linear Schrödinger family of equations. J Phys A: Math Gen, 1987, 20(8): 2003-2024 doi: 10.1088/0305-4470/20/8/020
[17]	Faddeev L D, Takhtajan L A. Hamiltonian Methods in the Theory of Solitons. Berlin: Springer, 1987