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

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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. 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 Chunjiang1,*(),Zhang Jian2()   

  1. 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:

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