中国物理B ›› 2022, Vol. 31 ›› Issue (11): 110201-110201.doi: 10.1088/1674-1056/ac7a1b

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Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

Guofei Zhang(张国飞)1, Jingsong He(贺劲松)2,†, and Yi Cheng(程艺)1   

  1. 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
    2 Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
  • 收稿日期:2022-05-09 修回日期:2022-05-09 接受日期:2022-06-18 出版日期:2022-10-17 发布日期:2022-10-17
  • 通讯作者: Jingsong He E-mail:hejingsong@szu.edu.cn
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 12071304 and 11871446) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515012554).

Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

Guofei Zhang(张国飞)1, Jingsong He(贺劲松)2,†, and Yi Cheng(程艺)1   

  1. 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
    2 Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
  • Received:2022-05-09 Revised:2022-05-09 Accepted:2022-06-18 Online:2022-10-17 Published:2022-10-17
  • Contact: Jingsong He E-mail:hejingsong@szu.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 12071304 and 11871446) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515012554).

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摘要： We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.

关键词: Schrödinger-type equation, Riemann-Hilbert problem, zero boundary conditions, N double-pole solitons

Abstract: We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.

Key words: Schrödinger-type equation, Riemann-Hilbert problem, zero boundary conditions, N double-pole solitons

中图分类号:  (Integrable systems)

  • 02.30.Ik
05.45.Yv (Solitons)