Skip to main content
SpringerLink
Log in

Mixed Higher-Order Rogue Waves and Solitons for the Coupled Modified Nonlinear Schrödinger Equation

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

The coupled modified nonlinear Schrödinger equation, which appears in birefringent optical fibers and describes the propagation of the short pluses in picosecond or femtosecond regions, is researched by the Darboux transformation. Utilizing both the Darboux transformation and the received special vector solution of the Lax pair, the general solutions for the coupled modified nonlinear Schrödinger equation are generated. Through searching the double-root condition of the spectral characteristic equation for the matrix in the Lax pair, we can obtain the reduced solutions that mixed higher-order rogue waves and solitons. Besides, the reduced solutions are mainly discussed in the following two types: (1) one component includes higher-order rogue waves and multi-bright-solitons, the other one is higher-order rogue waves and multi-dark-solitons; (2) higher-order rogue waves as the degenerate case. The dynamical behaviors of these reduced solutions are detailedly discussed.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Chen, S.H., Baronio, F., Soto-Crespo, J.M., Grelu, P., Mihalache, D.: Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems. J. Phys. A Math. Theor. 50, 463001 (2017)

    Article  MATH  Google Scholar 

  2. Li, R.M., Geng, X.G.: On a vector long wave-short wave-type model. Stud. Appl. Math. 144, 1–21 (2019)

    Google Scholar 

  3. Zhang, G.Q., Ling, L.M., Yan, Z.Y.: Multi-component nonlinear Schrödinger equations with nonzero boundary conditions: higher-order vector Peregrine solitons and asymptotic estimates. J. Nonlinear Sci. 31, 81 (2021)

    Article  MATH  Google Scholar 

  4. Akhmediev, N.: Waves that appear from nowhere: complex rogue wave structures and their elementary particles. Front. Phys. 8, 612318 (2021)

    Article  Google Scholar 

  5. Mihalache, D.: Localized structures in optical and matter-wave media: a selection of recent studies. Rom. Rep. Phys. 73, 403 (2021)

    Google Scholar 

  6. Ling, L.M., Guo, B.L., Zhao, L.C.: High-order rogue waves in vector nonlinear Schrödinger equations. Phys. Rev. E 89, 041201(R) (2014)

    Article  Google Scholar 

  7. Weng, W.F., Zhang, G.Q., Wang, L., Zhang, M.H., Yan, Z.Y.: Rational vector rogue waves for the n-component Hirota equation with non-zero backgrounds. Physica D 427, 133005 (2021)

    Article  MATH  Google Scholar 

  8. Li, B.Q., Ma, Y.L.: Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems. Chaos. Soliton. Fract. 156, 111832 (2022)

    Article  Google Scholar 

  9. Li, B.Q., Ma, Y.L.: A ‘firewall’ effect during the rogue wave and breather interactions to the Manakov system. Nonlinear Dyn. (2022). https://doi.org/10.1007/s11071-022-07878-6

    Article  Google Scholar 

  10. Guan, W.Y., Li, B.Q.: Asymmetrical and self-similar structures of optical breathers for the Manakov system in photorefractive crystals and randomly birefringent fibers. Optik 194, 162882 (2019)

    Article  Google Scholar 

  11. Sinthuja, N., Manikandan, K., Senthilvelan, M.: Formation of rogue waves on the periodic background in a fifth-order nonlinear Schrödinger equation. Phys. Lett. A 415, 127640 (2021)

    Article  MATH  Google Scholar 

  12. Manikandan, K., Vishnu Priya, N., Senthilvelan, M., Sankaranarayanan, R.: Higher-order matter rogue waves and their deformations in two-component Bose-Einstein condensates. Wave Random Complex 32, 867 (2022)

    Article  MATH  Google Scholar 

  13. Sinthuja, N., Manikandan, K., Senthilvelan, M.: Rogue waves on an elliptic function background in complex modified Korteweg-de Vries equation. Phys. Scripta 96, 105206 (2021)

    Article  MATH  Google Scholar 

  14. Sinthuja, N., Manikandan, K., Senthilvelana, M.: Rogue waves on the double-periodic background in Hirota equation. Eur. Phys. J. Plus 136, 305 (2021)

    Article  Google Scholar 

  15. Manikandan, K., Senthilvelan, M.: An analysis of spatiotemporal localized solutions in the variable coefficients (\(3+1\))-dimensional nonlinear Schrödinger equation with six different forms of dispersion parameters. Chaos 26, 073116 (2016)

    Article  MATH  Google Scholar 

  16. Li, B.Q., Ma, Y.L.: Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation. Appl. Math. Comput. 386, 125469 (2020)

    MATH  Google Scholar 

  17. Li, B.Q., Ma, Y.L.: Interaction dynamics of hybrid solitons and breathers for extended generalization of Vakhnenko equation. Nonlinear Dyn. 102, 1787–1799 (2020)

    Article  Google Scholar 

  18. Li, B.Q.: Hybrid breather and rogue wave solution for a (\(2+1\))-dimensional ferromagnetic spin chain system with variable coefficients. Int. J. Comput. Math. 99, 506 (2022)

    Article  MATH  Google Scholar 

  19. Ma, Y.L., Li, B.Q.: Hybrid rogue wave and breather solutions for a complex mKdV equation in few-cycle ultra-short pulse optics. Eur. Phys. J. Plus 137, 861 (2022)

    Article  Google Scholar 

  20. Ma, Y.L., Wazwaz, A.M., Li, B.Q.: New extended Kadomtsev-Petviashvili equation: multiple soliton solutions, breather, lump and interaction solutions. Nonlinear Dyn. 104, 1581–1594 (2021)

    Article  Google Scholar 

  21. Feng, B.F.: General N-soliton solution to a vector nonlinear Schrödinger equation. J. Phys. A: Math. Theor. 47, 355203 (2014)

    Article  MATH  Google Scholar 

  22. Chen, J.C., Chen, Y., Feng, B.F., Maruno, K.: General mixed multi-soliton solutions to one-dimensional multi-component Yajima-Oikawa system. J. Phys. Soc. Jpn. 84, 074001 (2015)

    Article  Google Scholar 

  23. Han, Z., Chen, Y., Chen, J.C.: Bright-dark mixed N-soliton solutions of the multi-component Mel’nikov system. J. Phys. Soc. Jpn. 86, 104008 (2017)

    Article  Google Scholar 

  24. Baronio, F., Degasperis, A., Conforti, M., Wabnitz, S.: Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109, 044102 (2012)

    Article  Google Scholar 

  25. Ling, L.M., Zhao, L.C.: Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simulat. 72, 449–471 (2019)

    Article  MATH  Google Scholar 

  26. Zhang, G.Q., Yan, Z.Y.: Three-component nonlinear Schrödinger equations: modulational instability, Nth-order vector rational and semi-rational rogue waves, and dynamics. Commun. Nonlinear Sci. Numer. Simul. 62, 117–133 (2018)

    Article  MATH  Google Scholar 

  27. Rao, J.G., Porsezian, K., He, J.S., Kanna, T.: Dynamics of lumps and dark-dark solitons in the multi-component long-wave-short-wave resonance interaction system. Proc. R. Soc. A 474, 20170627 (2018)

    Article  MATH  Google Scholar 

  28. Zhao, L.C., Liu, J.: Localized nonlinear waves in a two-mode nonlinear fiber. J. Opt. Soc. Amer. B 29, 3119C3127 (2012)

    Article  Google Scholar 

  29. Lin, S.N., Chen, Y.: A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions. J. Comput. Phys. 457, 111053 (2022)

    Article  MATH  Google Scholar 

  30. Xu, T., Chen, Y.: Mixed interactions of localized waves in the three-component coupled derivative nonlinear Schrödinger equations. Nonlinear Dynam. 92, 2133–2142 (2018)

    Article  Google Scholar 

  31. Chen, S.H.: Twisted rogue-wave pairs in the Sasa-Satsuma equation. Phys. Rev. E 88, 023202 (2013)

    Article  Google Scholar 

  32. Ye, Y.L., Zhou, Y., Chen, S.H., Baronio, F., Grelu, P.: General rogue wave solutions of the coupled Fokas-Lenells equations and non-recursive Darboux transformation. Proc. R. Soc. A 475(2224), 20180806 (2019)

    Article  MATH  Google Scholar 

  33. Xu, T., Chan, W.H., Chen, Y.: Higher-order rogue wave pairs in the coupled cubic-quintic nonlinear Schrödinger equations. Commun. Theor. Phys. 70, 153–160 (2018)

    Article  MATH  Google Scholar 

  34. Zhao, L.C., Liu, J.: Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation. Phys. Rev. E 87, 013201 (2013)

    Article  Google Scholar 

  35. Chen, J.B., Pelinovsky, D.E.: Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background. J. Nonlinear Sci. 29, 2797–2843 (2019)

    Article  MATH  Google Scholar 

  36. Feng, B.F., Ling, L.M., Takahashi, D.A.: Multi-breather and high-order rogue waves for the nonlinear Schrödinger equation on the elliptic function background. Stud. Appl. Math. 144, 46–101 (2020)

    Article  MATH  Google Scholar 

  37. Mu, G., Qin, Z.Y., Grimshaw, R.: Dynamics of rogue waves on a multi-soliton background in a vector nonlinear Schrödinger equation. SIAM J. Appl. Math. 75, 1–20 (2015)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  39. Hisakado, M., Wadati, M.: Integrable multi-componet hybrid nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 64, 408 (1995)

    Article  MATH  Google Scholar 

  40. 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, 4315–4321 (2009)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  42. 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, 279–289 (2010)

    Article  Google Scholar 

  43. Wadati, M., Konno, K., Ichikawa, Y.H.: A generalization of inverse scattering method. J. Phys. Soc. Jpn. 46, 1965–1966 (1979)

    Article  MATH  Google Scholar 

  44. He, J.S., Xu, S.W., Cheng, Y.: The rational solutions of the mixed nonlinear Schrödinger equation. AIP Adv. 5, 017105 (2015)

    Article  Google Scholar 

  45. W. B. Li, C. Y. Xue, L. L. Sun, The general mixed nonlinear Schrödinger equation: darboux transformation, rogue wave solutions, and modulation instability, Adv. Differ. Equ. 233 (2016)

  46. Wu, Q.L., Zhang, H.Q.: Breathers, rogue waves and breather-rogue waves on a periodic background for the modified nonlinear Schrödinger equation. Wave Motion 110, 102890 (2022)

    Article  MATH  Google Scholar 

  47. Chowdhury, A.R., Paul, S., Sen, S.: Periodic solutions of the mixed nonlinear Schrödinger equation. Phys. Rev. D 32, 12 (1985)

    Article  Google Scholar 

  48. Matsuno, Y.: The N-soliton solution of a two-component modified nonlinear Schrödinger equation. Phys. Lett. A 375, 3090–3094 (2011)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  51. Matsuno, Y.: The bright N-soliton solution of a multi-component modified nonlinear Schrödinger equation. J. Phys. A: Math. Theor. 44, 495202 (2011)

    Article  MATH  Google Scholar 

  52. Matsuno, Y.: The multi-component modified nonlinear Schrödinger system with nonzero boundary conditions. Phys. Scr. 94, 115216 (2019)

    Article  Google Scholar 

  53. 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  MATH  Google Scholar 

  54. Xu, T., He, G.L.: The coupled derivative nonlinear Schrödinger equation: conservation laws, modulation instability and semirational solutions. Nonlinear Dyn. 100, 2823–2837 (2020)

    Article  Google Scholar 

  55. Baronio, F., Conforti, M., Degasperis, A., Lombardo, S., Onorato, M., Wabnitz, S.: Vector rogue waves and baseband modulation instability in the defocusing regime. Phys. Rev. Lett. 113, 034101 (2014)

    Article  Google Scholar 

  56. Baronio, F., Chen, S., Grelu, P., Wabnitz, S., Conforti, M.: Baseband modulation instability as the origin of rogue waves. Phys. Rev. A 91, 033804 (2015)

    Article  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No.12201578, 11871232) and Natural Science Foundation of Henan Province (Grant No.222300420377, 212300410417), the Doctor Scientific Research Fund of Zhengzhou University of Light Industry and the Youth Core Teachers Foundation of Zhengzhou University of Light Industry.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, Henan, China

    Tao Xu, Guoliang He, Ming Wang & Yanqing Wang

Authors
  1. Tao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoliang He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ming Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yanqing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Xu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, T., He, G., Wang, M. et al. Mixed Higher-Order Rogue Waves and Solitons for the Coupled Modified Nonlinear Schrödinger Equation. Qual. Theory Dyn. Syst. 22, 14 (2023). https://doi.org/10.1007/s12346-022-00704-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00704-9

Keywords

Search

Navigation