Skip to main content
SpringerLink
Log in

Revised Riemann–Hilbert problem for the derivative nonlinear Schrödinger equation: Vanishing boundary condition

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

With a vanishing boundary condition, we consider a revised Riemann–Hilbert problem (RHP) for the derivative nonlinear Schrödinger equation (DNLS), where an integral factor is introduced such that the RHP satisfies the normalization condition. In the reflectionless situation, we construct the formulas for the \(N\)th-order solutions of the DNLS equation, including the solitons and positons that respectively correspond to \(N\) pairs of simple poles and one pair of \(N\)th-order poles of the RHP. According to the Cauchy–Binet formula, we show the expressions for \(N\)th-order solitons. Additionally, we give an explicit expression for the second-order positon and graphically describe evolutions of the third-order and fourth-order positons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.

Similar content being viewed by others

References

  1. A. Rogister, “Parallel propagation of nonlinear low-frequency waves in high-\(\beta\) plasma,” Phys. Fluids, 14, 2733–2739 (1971).

    Article  ADS  Google Scholar 

  2. K. Mio, T. Ogino, K. Minami, and S. Takeda, “Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas,” J. Phys. Soc. Japan, 41, 265–271 (1976).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. E. Mjølhus, “On the modulational instability of hydromagnetic waves parallel to the magnetic field,” J. Plasma Phys., 16, 321–334 (1976).

    Article  ADS  Google Scholar 

  4. E. Mjølhus and T. Hada, “Soliton theory of quasi-parallel MHD waves,” in: Nonlinear Waves and Chaos in Space Plasmas (T. Hada and H. Matsumoto, eds.), Terra Sci., Tokyo (1977), pp. 121–169.

    Google Scholar 

  5. E. Mjølhus, “Nonlinear Alfvén waves and the DNLS equation: Oblique aspects,” Phys. Scr., 40, 227–237 (1989).

    Article  ADS  Google Scholar 

  6. S. R. Spangler, “Nonlinear evolution of MHD waves at the Earth’s bow shock,” in: Nonlinear Waves and Chaos in Space Plasmas (T. Hada and H. Matsumoto, eds.), Terra Sci., Tokyo (1977), pp. 171–224.

    Google Scholar 

  7. K. Baumgärtel, “Soliton approach to magnetic holes,” J. Geophys. Res., 104, 28295–28308 (1999).

    Article  ADS  Google Scholar 

  8. C. F. Kennel, B. Buti, T. Hada, and R. Pellat, “Nonlinear, dispersive, elliptically polarized Alfvén waves,” Phys. Fluids, 31, 1949–1961 (1988).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  9. D. J. Kaup and A. C. Newell, “An exact solution for a derivative nonlinear Schrödinger equation,” J. Math. Phys., 19, 798–801 (1978).

    Article  MATH  ADS  Google Scholar 

  10. G.-Q. Zhou and N.-N. Huang, “An \(N\)-soliton solution to the DNLS equation based on revised inverse scattering transform,” J. Phys. A: Math. Theor., 40, 13607–13623 (2007).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. C.-N. Yang, J.-L. Yu, H. Cai, and N.-N. Huang, “Inverse scattering transform for the derivative nonlinear Schrödinger equation,” Chinese Phys. Lett., 25, 421–424 (2008).

    Article  ADS  Google Scholar 

  12. G. Zhang and Z. Yan, “The derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions: Inverse scattering transforms and \(N\)-double-pole solutions,” J. Nonlinear Sci., 30, 3089–3127 (2020).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. Y. Zhang, X. Tao, and S. Xu, “The bound-state soliton solutions of the complex modified KdV equation,” Inverse Problems, 36, 065003, 17 pp. (2020).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. H. Steudel, “The hierarchy of multi-soliton solutions of the derivative nonlinear Schrödinger equation,” J. Phys. A: Math. Gen., 36, 1931–1946 (2003).

    Article  MATH  ADS  Google Scholar 

  15. S. W. Xu, J. S. He, and L. H. Wang, “The Darboux transformation of the derivative nonlinear Schrödinger equation,” J. Phys. A: Math. Theor., 44, 305203, 22 pp. (2011).

    Article  MATH  Google Scholar 

  16. Y. Zhang, D. Qiu, and J. He, “Explicit \(N\)th order solutions of Fokas–Lenells equation based on revised Riemann–Hilbert approach,” J. Math. Phys., 64, 053502, 14 pp. (2023).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Y. Xiao and E. Fan, “A Riemann–Hilbert approach to the Harry-Dym equation on the line,” Chinese Ann. Math. Ser. B, 37, 373–384 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Zhang, J. Rao, Y. Cheng, and J. He, “Riemann–Hilbert method for the Wadati–Konno–Ichikawa equation: \(N\) simple poles and one higher-order pole,” Phys. D, 399, 173–185 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Ai and J. Xu, “On a Riemann–Hilbert problem for the Fokas–Lenells equation,” Appl. Math. Lett., 87, 57–63 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  20. X. Ma, “Riemann–Hilbert approach for a higher-order Chen–Lee–Liu equation with high- order poles,” Commun. Nonlinear Sci. Numer. Simul., 114, 106606, 14 pp. (2022).

    Article  MathSciNet  MATH  Google Scholar 

  21. Z.-X. Zhou, “Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul., 62, 480–488 (2018).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. Xinxin Ma and Yonghui Kuang, “Inverse scattering transform for a nonlocal derivative nonlinear Schrödinger equation,” Theoret. and Math. Phys., 210, 31–45 (2022).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  23. M. J. Ablowitz, X.-D. Luo, Z. H. Musslimani, and Y. Zhu, “Integrable nonlocal derivative nonlinear Schrödinger equations,” Inverse Problems, 38, 065003, 34 pp. (2022).

    Article  MATH  ADS  Google Scholar 

  24. J. Lenells, “The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line,” Phys. D, 240, 512–525 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  25. A. S. Fokas and J. Lenells, “The unified method: I. Nonlinearizable problems on the half-line,” J. Phys. A: Math. Theor., 45, 195201, 38 pp. (2012).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  26. J. Lenells and A. S. Fokas, “The unified method: II. NLS on the half-line with \(t\)-periodic boundary conditions,” J. Phys. A: Math. Theor., 45, 195202, 36 pp. (2012); “The unified method: III. Nonlinearizable problems on the interval,” 195203, 21 pp.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  27. B.-B. Hu, T.-C. Xia, N. Zhang, and J.-B. Wang, “Initial-boundary value problems for the coupled higher-order nonlinear Schrödinger equations on the half-line,” Internat. J. Nonlinear Numer. Simul., 19, 83–92 (2018).

    Article  MATH  Google Scholar 

  28. B. Hu, L. Zhang, T. Xia, and Z. Ning, “On the Riemann–Hilbert problem of the Kundu equation,” Appl. Math. Comput., 381, 125262, 14 pp. (2020).

    MathSciNet  MATH  Google Scholar 

  29. B. Hu, L. Zhang, and N. Zhang, “On the Riemann–Hilbert problem for the mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation,” J. Comput. Appl. Math., 390, 113393, 14 pp. (2021).

    Article  MATH  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China (grant No. 12171433) and the Doctoral research foundation project of Huizhou University (grant No. 2022JB039).

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, Zhejiang province, China

    Yongshuai Zhang & Haibing Wu

  2. School of Mathematics and Statistics, Huizhou University, Huizhou, Guangdong province, China

    Deqin Qiu

Authors
  1. Yongshuai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haibing Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Deqin Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongshuai Zhang.

Ethics declarations

The authors declare no conflicts of interest.

Additional information

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 204–219 https://doi.org/10.4213/tmf10517.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Wu, H. & Qiu, D. Revised Riemann–Hilbert problem for the derivative nonlinear Schrödinger equation: Vanishing boundary condition. Theor Math Phys 217, 1595–1608 (2023). https://doi.org/10.1134/S0040577923100112

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577923100112

Keywords

Search

Navigation