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

Acta mathematica scientia,Series A ›› 2023, Vol. 43 ›› Issue (1): 101-122.

Previous Articles Next Articles

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)

Abstract

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

CLC Number:

O175

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.

Trendmd

References

[1]	Kundu A. Landau-Lifshitz and higherorder nonlinear systems gauge generated from nonlinear Schr?dinger type equations. J Math Phys, 1984, 25(12): 3433-3438
[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 Sch?dinger equation for application to deep water waves. Proc R Soc Lond A, 1979, 369(1736): 105-114
[4]	Chan H N, Chow K W, Kedziora D J. Rogue wave modes for a derivative nonlinear Schr?dinger model. Phys Rev E, 2014, 89(3): 032914
[5]	Hu B B, Zhang L, Zhang N. On the Riemann-Hilbert problem for the mixed Chen-Lee-Liu derivative nonlinear Schr?dinger equation. J Comput Appl Math, 2021, 390: 113393
[6]	Zhang Y S, Guo L J, Chabchoub A. Higher-order rogue wave dynamics for a derivative nonlinear Schr?dinger 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 Schr?dinger 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
[9]	Biondini G, Kova?i? G. Inverse scattering transform for the focusing nonlinear Schr?dinger equation with nonzero boundary conditions. J Math Phys, 2014, 55(3): 031506
[10]	Pichler M, Biondini G. On the focusing non-linear Schr?dinger equation with non-zero boundary conditions and double poles. IMA J Appl Math, 2017, 82(1): 131-151
[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 Schr?dinger-type equation with nonzero boundary conditions. Modern Phys Lett B, 2021, 35(12): 2150208
[13]	Zhang G Q, Yan Z Z. The derivative nonlinear Schr?dinger equation with zero/nonzero boundary conditions: Inverse scattering transforms and N-double-pole solutions. J Nonlinear Sci, 2020, 30(2): 3089-3127
[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
[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
[16]	Clarkson P A, Cosgrove C M. Painlevé analysis of the non-linear Schr?dinger family of equations. J Phys A: Math Gen, 1987, 20(8): 2003-2024
[17]	Faddeev L D, Takhtajan L A. Hamiltonian Methods in the Theory of Solitons. Berlin: Springer, 1987

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

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 5

Metrics

Comments

Recommended 0

[1]	Lou Yu, Zhang Yi. Breather and Rogue Wave on the Periodic/Double Periodic Background and Interaction Solutions of the Generalized Derivative Nonlinear Schr $\rm\ddot{o}$ dinger Equation [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1511-1519.
[2]	Jian Hui, Gong Min, Wang Li. On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement [J]. Acta mathematica scientia,Series A, 2023, 43(5): 1350-1372.
[3]	Wang Guixian,Wang XiuBin,Han Bo. Inverse Scattering Transform for the Focusing Kundu-Eckhaus Equation: Long-time Dynamics of the Steplike Oscillating Background [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1085-1122.
[4]	Li Deke, Wang Qingxuan. Limit Behavior of Mass Critical Inhomogeneous Schrödinger Equation at the Threshold [J]. Acta mathematica scientia,Series A, 2023, 43(1): 123-131.
[5]	Beibei Hu,Ling Zhang,Ning Zhang. An Initial-Boundary Value Problem for the Higher-Order Chen-Lee-Liu Equation on the Half-Line [J]. Acta mathematica scientia,Series A, 2020, 40(2): 422-431.