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The breather and semi-rational rogue wave solutions for the coupled mixed derivative nonlinear Schrödinger equations

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Abstract

The coupled mixed derivative nonlinear Schrödinger equations, correlated with Lax pairsinvolving \(3\times 3\) matrices, arise as a significant integrable system in many physical contexts. By constructing the Darboux transformation, breathing bright–dark solitons, mixed kink solutions, mixed periodic solutions, semi-rational rogue wave solutions and various types of mixed soliton solutions are attained. Furthermore, breather fusion, breather fission and higher-order solutions are derived, whose dynamic behaviors are illustrated graphically to distinguish different parameter values.

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The data that support the findings of this article are available from the corresponding author upon reasonable request.

References

  1. Li, M., Xiao, J.H., Liu, W.J., Wang, P., Qin, B., Tian, B.: Mixed-type vector solitons of the \(N\)-coupled mixed derivative nonlinear Schrödinger equations from optical fibers. Phys. Rev. E 87(3), 032914 (2013)

    Article  Google Scholar 

  2. Li, M., Xiao, J.H., Qin, B., Wang, M., Tian, B.: Vector-soliton bound states for the coupled mixed derivative nonlinear Schrödinger equations in optical fibers. Wave Motion 50(1), 1–10 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Janutka, A.: Collisions of optical ultra-short vector pulses. J. Phys. A 41(28), 285204 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Li, M., Tian, B., Liu, W.J., Jiang, Y., Sun, K.: Dark and anti-dark vector solitons of the coupled modified nonlinear Schrödinger equations from the birefringent optical fibers. Eur. Phys. J. D 59(2), 279–289 (2010)

    Article  Google Scholar 

  5. Li, M., Xiao, J.H., Jiang, Y., Wang, M., Tian, B.: Bound-state dark/antidark solitons for the coupled mixed derivative nonlinear Schrödinger equations in optical fibers. Eur. Phys. J. D 66(11), 1–14 (2012)

    Article  Google Scholar 

  6. Zhang, H.Q., Tian, B., Lü, X., Li, H., Meng, X.H.: Soliton interaction in the coupled mixed derivative nonlinear Schrödinger equations. Phys. Lett. A 373(47), 4315–4321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Porsezian, K.: Soliton models in resonant and nonresonant optical fibers. Pramana J. Phys. 57(5), 1003–1039 (2001)

    Article  Google Scholar 

  8. Geng, X.G., Li, R.M., Xue, B.: A vector general nonlinear Schrödinger equation with \((m+n)\) components. J. Nonlinear Sci. 30, 991 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger-type equation. J. Math. Phys. 19, 798 (1978)

    Article  MATH  Google Scholar 

  10. Zhang, Y., Yang, J.W.: Solitons, breathers and rogue waves for the coupled Fokas-Lenells system via Darboux transformation. Nonlinear Anal-Real. 33, 237–252 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang, L., Liu, C., Wu, X., Wang, X., Sun, W.R.: Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation. Nonlinear Dyn. 94, 977 (2018)

    Article  Google Scholar 

  12. Guo, R., Hao, H.Q., Zhang, L.L.: Dynamic behaviors of the breather solutions for the AB system in fluid mechanics. Nonlinear Dyn. 74(3), 701–709 (2013)

    Article  MathSciNet  Google Scholar 

  13. Ji, T., Zhai, Y.: Soliton, breather and rogue wave solutions of the coupled Gerdjikov-Ivanov equation via Darboux transformation. Nonlinear Dyn. 101(1), 619–631 (2020)

    Article  Google Scholar 

  14. Tajiri, M., Watanabe, Y.: Breather solutions to the focusing nonlinear Schrödinger equation. Phys. Rev. E 57(3), 3510 (1998)

    Article  MathSciNet  Google Scholar 

  15. Kivshar, Y.S., Flach, S.: Focus issue: nonlinear localized modes: physics and applications. Chaos 13, 586 (2003)

    Article  Google Scholar 

  16. Xie, X.Y., Yang, S.K., Ai, C.H., Kong, L.C.: Integrable turbulence for a coupled nonlinear Schrödinger system. Phys. Lett. A 384, 126119 (2020)

    Article  Google Scholar 

  17. Chen, J.B., Pelinovsky, D.E.: Rogue periodic waves of the focusing nonlinear Schrödinger equation. Proc. R. Soc. A 474, 20170814 (2018)

    Article  MATH  Google Scholar 

  18. Frisquet, B., Kibler, B., Morin, P., Baronio, F., Conforti, M., Millot, G., Wabnitz, S.: Optical dark rogue wave. Sci. Rep. 6, 20785 (2016)

    Article  Google Scholar 

  19. Ankiewicz, A., Wang, Y., Wabnitz, S., Akhmediev, N.: Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions. Phys. Rev. E 89(1), 012907 (2014)

    Article  Google Scholar 

  20. Hisakado, M., Iizuka, T., Wadati, M.: Coupled hybrid nonlinear Schrödinger equation and optical solitons. J. Phys. Soc. Jpn. 63(8), 2887–2894 (1994)

    Article  Google Scholar 

  21. Hisakado, M., Wadati, M.: Integrable multi-component hybrid nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 64(2), 408–413 (1995)

    Article  MATH  Google Scholar 

  22. Wu, R., Jiang, W., Li, L.: Homoclinic orbits for coupled modified nonlinear Schrödinger equations. Chaos, Solitons & Fractals 38(4), 1093–1103 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, H.Q.: Darboux transformation and \(N\)-soliton solution for the coupled modified nonlinear Schrödinger equations. Z. Naturforsch. A 67(12), 711–722 (2012)

    Article  Google Scholar 

  24. Zhang, H.Q.: Energy-exchange collisions of vector solitons in the \(N\)-coupled mixed derivative nonlinear Schrödinger equations from the birefringent optical fibers. Opt. Commun. 290, 141–145 (2013)

    Article  Google Scholar 

  25. Hu, B., Xia, T.: A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half-line. Math. Methods Appl. Sci. 41(13), 5112–5123 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yan, X.W.: Lax pair, Darboux-dressing transformation and localized waves of the coupled mixed derivative nonlinear Schrödinger equation in a birefringent optical fiber. Appl. Math. Lett. 107, 106414 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hang, C., Wu, Q.L., Zhang, H.Q.: Breathers and double-pole solutions of coupled mixed derivative nonlinear Schrödinger equations from optical fibers. Mod. Phys. Lett. B 35(22), 2150373 (2021)

    Article  Google Scholar 

  28. Song, N., Lei, Y., Cao, D.: Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrödinger equation. Acta Mech. Sinica 38(5), 1–7 (2022)

    Article  Google Scholar 

  29. Song, N., Lei, Y.X., Zhang, Y.F., Zhang, W.: Localized waves for the coupled mixed derivative nonlinear Schrödinger equation in a birefringent optical fiber. J. Nonlinear Math. Phys. 29, 318–330 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 86, 5043 (2001)

    Article  Google Scholar 

  31. Degasperis, A., Lombardo, S.: Multicomponent integrable wave equations II. Soliton solutions. J. Phys. A 42, 385206 (2009)

  32. Kaup, D.J., Malomed, B.A., Tasgal, R.S.: Internal dynamics of a vector soliton in a nonlinear optical fiber. Phys. Rev. E 48, 3049 (1993)

    Article  MathSciNet  Google Scholar 

  33. Yin, H.M., Tian, B., Zhao, X.C.: Chaotic breathers and breather fission/fusion for a vector nonlinear Schrödinger equation in a birefringent optical fiber or wavelength division multiplexed system. Comput. Math. Appl. 368, 124768 (2020)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11371326 and Grant No. 11975145).

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11371326 and Grant No. 11975145).

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Jie Jin, Yi Zhang, Rusuo Ye & Lifei Wu

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  1. Jie Jin
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Correspondence to Yi Zhang.

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Jin, J., Zhang, Y., Ye, R. et al. The breather and semi-rational rogue wave solutions for the coupled mixed derivative nonlinear Schrödinger equations. Nonlinear Dyn 111, 633–643 (2023). https://doi.org/10.1007/s11071-022-07834-4

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