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Inverse scattering transform for a nonlocal derivative nonlinear Schrödinger equation

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Abstract

We give a detailed discussion of a nonlocal derivative nonlinear Schrödinger (NL-DNLS) equation with zero boundary conditions at infinity in terms of the inverse scattering transform. The direct scattering problem involves discussions of the analyticity, symmetries, and asymptotic behavior of the Jost solutions and scattering coefficients, and the distribution of the discrete spectrum points. Because of the symmetries of the NL-DNLS equation, the discrete spectrum is different from those for DNLS-type equations. The inverse scattering problem is solved by the method of a matrix Riemann–Hilbert problem. The reconstruction formula, the trace formula, and explicit solutions are presented. The soliton solutions with special parameters for the NL-DNLS equation with a reflectionless potential are obtained, which may have singularities.

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References

  1. V. S. Gerdjikov, G. Vilasi, and A. B. Yanovski, Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods (Lecture Notes in Physics, Vol. 748), Springer, Berlin, Heidelberg (2008).

    Book  Google Scholar 

  2. W.-X. Ma and Y. You, “Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions,” Trans. Amer. Math. Soc., 357, 1753–1778 (2005).

    Article  MathSciNet  Google Scholar 

  3. X.-R. Hu, S.-Y. Lou, and Y. Chen, “Explicit solutions from eigenfunction symmetry of the Korteweg–de Vries equation,” Phys. Rev. E, 85, 056607, 8 pp. (2012).

    Article  ADS  Google Scholar 

  4. V. N. Serkin and A. Hasegava, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett., 85, 4502–4505 (2000).

    Article  ADS  Google Scholar 

  5. B. Guo, L. Ling, and Q. P. Liu, “Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions,” Phys. Rev. E, 85, 026607, 9 pp. (2012).

    Article  ADS  Google Scholar 

  6. P. Felmer, A. Quaas, and J. Tan, “Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian,” Proc. Roy Soc. Edinburgh Sect. A, 142, 1237–1262 (2012).

    Article  MathSciNet  Google Scholar 

  7. D. J. Benney and A. C. Newell, “Propagation of nonlinear wave envelopes,” J. Math. Phys., 46, 133–139 (1967).

    Article  MathSciNet  Google Scholar 

  8. M. J. Ablowitz and Z. H. Musslimani, “Integrable nonlocal nonlinear Schrödinger equation,” Phys. Rev. Lett., 110, 064105, 5 pp. (2013).

    Article  ADS  Google Scholar 

  9. M. J. Ablowitz and Z. H. Musslimani, “Integrable discrete PT symmetric model,” Phys. Rev. E, 90, 032912, 5 pp. (2014).

    Article  ADS  Google Scholar 

  10. M. J. Ablowitz and Z. H. Musslimani, “Integrable nonlocal nonlinear equations,” Stud. Appl. Math., 139, 7–59 (2017).

    Article  MathSciNet  Google Scholar 

  11. J. Yang, “General \(N\)-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations,” Phys. Lett. A, 383, 328–337 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  12. B. Yang and J. Yang, “Transformations between nonlocal and local integrable equations,” Stud. Appl. Math., 140, 178–201 (2018).

    Article  MathSciNet  Google Scholar 

  13. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math., 53, 249–315 (1974).

    Article  MathSciNet  Google Scholar 

  14. V. S. Gerdjikov and A. Saxena, “Complete integrability of nonlocal nonlinear Schrödinger equation,” J. Math. Phys., 58, 013502, 33 pp. (2017).

    Article  ADS  MathSciNet  Google Scholar 

  15. M. J. Ablowitz and Z. H. Musslimani, “Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation,” Nonlinearity, 29, 915–946 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  16. 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  ADS  MathSciNet  Google Scholar 

  17. M. J. Ablowitz, G. Biondini, and B. Prinari, “Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions,” Inverse Problems, 23, 1711–1758 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  18. G. Biondini and G. Kovačič, “Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions,” J. Math. Phys., 55, 031506, 22 pp. (2014).

    Article  ADS  MathSciNet  Google Scholar 

  19. M. J. Ablowitz, X.-D. Luo, and Z. H. Musslimani, “Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions,” J. Math. Phys., 59, 011501, 42 pp. (2018).

    Article  ADS  MathSciNet  Google Scholar 

  20. B. Prinari, M. J. Ablowitz, and G. Biondini, “Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions,” J. Math. Phys., 47, 063508, 33 pp. (2006).

    Article  ADS  MathSciNet  Google Scholar 

  21. J.-L. Ji and Z.-N. Zhu, “Soliton solutions of an integrable nonlocal modified Korteweg–de Vries equation through inverse scattering transform,” J. Math. Anal. Appl., 453, 973–984 (2017).

    Article  MathSciNet  Google Scholar 

  22. J. Wu, “Riemann–Hilbert approach and nonlinear dynamics in the nonlocal defocusing nonlinear Schrödinger equation,” Eur. Phys. J. Plus, 135, 523, 13 pp. (2020).

    Article  Google Scholar 

  23. G. Biondini and D. Kraus, “Inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions,” SIAM J. Math. Anal., 47, 706–757 (2015).

    Article  MathSciNet  Google Scholar 

  24. B. Zhang and E. Fan, “Riemann–Hilbert approach for a Schrödinger-type equation with nonzero boundary conditions,” Modern Phys. Lett. B, 35, 2150208, 32 pp. (2021).

    Article  ADS  Google Scholar 

  25. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg–de Vries equation,” Phys. Rev. Lett., 19, 1095–1097 (1967).

    Article  ADS  Google Scholar 

  26. B. Guo and L. Ling, “Riemann–Hilbert approach and \(N\)-soliton formula for coupled derivative Schrödinger equation,” J. Math. Phys., 53, 073506, 20 pp. (2012).

    Article  ADS  MathSciNet  Google Scholar 

  27. D.-S. Wang and X. Wang, “Long-time asymptotics and the bright \(N\)-soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approach,” Nonlinear Anal. Real World Appl., 41, 334–361 (2018).

    Article  MathSciNet  Google Scholar 

  28. X. Geng and J. Wu, “Riemann–Hilbert approach and \(N\)-soliton solutions for a generalized Sasa–Satsuma equation,” Wave Motion, 60, 62–72 (2016).

    Article  MathSciNet  Google Scholar 

  29. Q. Cheng and E. Fan, “Long-time asymptotics for a mixed nonlinear Schrödinger equation with the Schwartz initial data,” J. Math. Anal. Appl., 489, 124188, 24 pp. (2020).

    Article  MathSciNet  Google Scholar 

  30. S. Chen and Z. Yan, “Long-time asymptotics of solutions for the coupled dispersive AB system with initial value problems,” J. Math. Anal. Appl., 498, 124966, 31 pp. (2021).

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  32. 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  ADS  MathSciNet  Google Scholar 

  33. G. Zhou, “A newly revised inverse scattering transform for DNLS\(^{+}\) equation under nonvanishing boundary condition,” Wuhan Univ. J. Nat. Sci., 17, 144–150 (2012).

    Article  MathSciNet  Google Scholar 

  34. V. M. Lashkin, “\(N\)-soliton solutions and perturbation theory for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions,” J. Phys. A: Math. Theor., 40, 6119–6132 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  35. X.-J. Chen and W. K. Lam, “Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions,” Phys. Rev. E, 69, 066604, 8 pp. (2004).

    Article  ADS  MathSciNet  Google Scholar 

  36. 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 

  37. Z.-X. Zhou, “Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul., 62, 480–488 (2018); ar**v: 1612.04892.

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgments

We thank G. Q. Zhang for the numerous useful discussions.

Funding

This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 11931017 and 12001560), the Yue Qi Young Scholar Project, China University of Mining and Technology, Bei**g (Grant No. 00-800015Z1201), and the Fundamental Research Funds for Central Universities (Grant No. 00-800015A566).

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Bei**g, China

    **nxin Ma

  2. College of Science, Zhongyuan University of Technology, Zhengzhou, China

    Yonghui Kuang

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  1. **nxin Ma
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  2. Yonghui Kuang
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Correspondence to **nxin Ma.

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The authors declare no conflicts of interest.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 210, pp. 38–53 https://doi.org/10.4213/tmf10150.

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Ma, X., Kuang, Y. Inverse scattering transform for a nonlocal derivative nonlinear Schrödinger equation. Theor Math Phys 210, 31–45 (2022). https://doi.org/10.1134/S0040577922010032

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  • DOI: https://doi.org/10.1134/S0040577922010032

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