Advertisement

Nonlinear Dynamics

, Volume 92, Issue 4, pp 2133–2142 | Cite as

Mixed interactions of localized waves in the three-component coupled derivative nonlinear Schrödinger equations

  • Tao Xu
  • Yong Chen
Original Paper

Abstract

The Darboux transformation of the three-component coupled derivative nonlinear Schrödinger equations is constructed. Based on the special vector solution generated from the corresponding Lax pair, various interactions of localized waves are derived. Here, we focus on the higher-order interactional solutions among higher-order rogue waves, multi-solitons, and multi-breathers. It is defined as the identical type of interactional solution that the same combination appears among these three components \(q_1, q_2\), and \(q_3\), without considering different arrangements among them. According to our method and definition, these interactional solutions are completely classified as six types, among which there are four mixed interactions of localized waves in these three different components. In particular, the free parameters \(\mu \) and \(\nu \) play the important roles in dynamics structures of the interactional solutions. For example, different nonlinear localized waves merge with each other by increasing the absolute values of these two parameters. Additionally, these results demonstrate that more abundant and novel localized waves may exist in the multi-component coupled systems than in the uncoupled ones.

Keywords

Interactions of localized waves Rogue wave Soliton Breather Three-component coupled derivative nonlinear Schrödinger equations Darboux transformation 

Notes

Acknowledgements

We are greatly indebted to the editor and reviewer for their helpful comments and constructive suggestions, and we also express our sincere thanks to Xiaoyan Tang, Xin Wang, and other members of our discussion group for their valuable comments.

Compliance with ethical standards

Conflicts of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. 1.
    Zabusky, N.J., Kruskal, M.D.: Interaction of ”solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240 (1965)CrossRefzbMATHGoogle Scholar
  2. 2.
    Hasegawa, A., Kodama, Y.: Solitons in Optical Communication. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  3. 3.
    Malomed, B., Torner, L., Wise, F., Mihalache, D.: On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics. J. Phys. B At. Mol. Opt. Phys. 49, 170502 (2016)CrossRefGoogle Scholar
  4. 4.
    Mihalache, D.: Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature. Rom. Rep. Phys. 69, 403 (2017)Google Scholar
  5. 5.
    Kevrekidis, P.G., Frantzeskakis, D.J.: Solitons in coupled nonlinear Schrodinger models: a survey of recent developments. Rev. Phys. 1, 140–153 (2016)CrossRefGoogle Scholar
  6. 6.
    Ohta, Y., Wang, D.S., Yang, J.K.: General N-Dark–Dark solitons in the coupled nonlinear Schrödinger equations. Stud. Appl. Math. 127, 345–371 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ling, L.M., Zhao, L.C., Guo, B.L.: Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations. Nonlinearity 28, 3243–61 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chowdury, A., Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions. Phys. Rev. E 91, 022919 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Soto-Crespo, J.M., Devine, N., Akhmediev, N.: Integrable turbulence and rogue waves: breathers or solitons? Phys. Rev. Lett. 116, 103901 (2016)CrossRefGoogle Scholar
  10. 10.
    Akhmediev, N., Soto-Crespo, J.M., Ankiewicz, A.: How to excite a rogue wave. Phys. Rev. A 80, 043818 (2009)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Ohta, Y., Yang, J.K.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A 468, 1716–40 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, L.H., He, J.S., Xu, H., Wang, J., Porsezian, K.: Generation of higher-order rogue waves from multibreathers by double degeneracy in an optical fiber. Phys. Rev. E 95, 042217 (2017)CrossRefGoogle Scholar
  14. 14.
    Yan, Z.Y.: Vector financial rogue waves. Phys. Lett. A 375, 4274–4279 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Akhmediev, N., Eleonskii, V.M., Kulagin, N.E.: Exact first-order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 72, 183–96 (1987)CrossRefGoogle Scholar
  16. 16.
    Akhmediev, N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69, 1089–93 (1986)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ma, Y.C.: The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60, 43–58 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B 25, 16–43 (1983)CrossRefzbMATHGoogle Scholar
  19. 19.
    Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–8 (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Garrett, C., Gemmrich, J.: Rogue waves. Phys. Today 62, 62–63 (2009)CrossRefGoogle Scholar
  21. 21.
    Chabchoub, A., Hoffmann, N.P., Akhmediev, N.: Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106, 204502 (2011)CrossRefGoogle Scholar
  22. 22.
    Ling, L.M., Zhao, L.C., Yang, Z.Y., Guo, B.L.: Generation mechanisms of fundamental rogue wave spatial-temporal structure. Phys. Rev. E 96, 022211 (2017)CrossRefGoogle Scholar
  23. 23.
    Wang, X., Liu, C., Wang, L.: Rogue waves and W-shaped solitons in the multiple self-induced transparency system. Chaos 27, 093106 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Bilman, D., Miller, P.D.: A robust inverse scattering transform for the focusing nonlinear Schrödinger equation. arXiv:1710.06568v1
  25. 25.
    Ankiewicz, A., Akhmediev, N.: Multi-rogue waves and triangular numbers. Rom. Rep. Phys. 69, 104 (2017)Google Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Rogue waves and solitons on a cnoidal background. Eur. Phys. J. Spec. Top. 223, 43–62 (2014)CrossRefGoogle Scholar
  28. 28.
    Chen, C.L., Lou, S.Y.: CTE solvability and exact solution to the Broer–Kaup system. Chin. Phys. Lett. 30, 110202 (2013)CrossRefGoogle Scholar
  29. 29.
    Lou, S.Y.: Consistent Riccati expansion for integrable systems. Stud. Appl. Math. 134, 372–402 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rao, J.G., Liu, Y.B., Qian, C., He, J.S.: Rogue waves and hybrid solutions of the Boussinesq equation. Z. Naturforsch. A 72, 307–314 (2017)Google Scholar
  31. 31.
    Rao, J.G., Porsezian, K., He, J.S.: Semi-rational solutions of the third-type Davey–Stewartson equation. Chaos 27, 083115 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rao, J.G., Cheng, Y., He, J.S.: Rational and semirational solutions of the nonlocal Davey–Stewartson equations. Stud. Appl. Math. 139, 568–598 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, X.N., Chen, Y.: Rogue wave and a pair of resonance stripe solitons to a reduce d (3+1)-dimensional Jimbo–Miwa equation. Commun. Nonlinear Sci. Numer. Simul. 52, 24–31 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, X.N., Chen, Y.: Deformation rogue wave to the (2+1)-dimensional KdV equation. Nonlinear Dyn. 90, 755–763 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Chen, J.C., Chen, Y., Feng, B.F., Maruno, K.: General mixed multi-soliton solutions to one-dimensional multicomponent Yajima–Oikawa system. J. Phys. Soc. Jpn. 84, 074001 (2015)CrossRefGoogle Scholar
  36. 36.
    Han, Z., Chen, Y.: General N-dark soliton solutions of the multi-component Mel’nikov system. J. Phys. Soc. Jpn. 86, 074005 (2017)CrossRefGoogle Scholar
  37. 37.
    Guo, B.L., Ling, L.M.: Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations. Chin. Pyhs. Lett. 28, 110202 (2011)CrossRefGoogle Scholar
  38. 38.
    Wang, X., Chen, Y.: Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations. Chin. Phys. B. 23, 070203 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhang, H.Q., Zhang, M.Y., Hu, R.: Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schrödinger equation. Appl. Math. Lett. 76, 170–174 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhang, Y., Liu, Y.P., Tang, X.Y.: A general integrable three-component coupled nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 89, 2729–2738 (2017)CrossRefGoogle Scholar
  41. 41.
    Zhang, G.Q., Yan, Z.Y., Wen, X.Y., Chen, Y.: Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations. Phys. Rev. E 95, 042201 (2017)CrossRefGoogle Scholar
  42. 42.
    Morris, H.C., Dodd, R.K.: The two component derivative nonlinear Schrödinger equation. Phys. Scr. 20, 505 (1979)CrossRefzbMATHGoogle Scholar
  43. 43.
    Ling, L.M., Liu, Q.P.: Darboux transformation for a two-component derivative nonlinear Schrödinger equation. J. Phys. A Math. Theor. 43, 434023 (2010)CrossRefzbMATHGoogle Scholar
  44. 44.
    Rogister, A.: Parallel propagation of nonlinear low-frequency waves in high-\(\beta \) plasma. Phys. Fluids 14, 2733 (1971)CrossRefGoogle Scholar
  45. 45.
    Mjøhus, E.: On the modulational instability of hydromagnetic waves parallel to the magnetic field. J. Plasma Phys 16, 321 (1976)CrossRefGoogle Scholar
  46. 46.
    Mjøhus, E.: Nonlinear Alfvén waves and the DNLS equation: oblique aspects. Phys. Scr. 40, 227 (1989)CrossRefGoogle Scholar
  47. 47.
    Ruderman, M.S.: DNLS equation for large-amplitude solitons propagating in an arbitrary direction in a high-\(\beta \) Hall plasma. J. Plasma Phys. 67, 271 (2002)CrossRefGoogle Scholar
  48. 48.
    Xu, S.W., He, J.S., Wang, L.H.: The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A Math. Theor. 44, 305203 (2011)CrossRefzbMATHGoogle Scholar
  49. 49.
    Guo, B.L., Ling, L.M., Liu, Q.P.: High-order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations. Stud. Appl. Math. 130, 317–344 (2012)CrossRefzbMATHGoogle Scholar
  50. 50.
    Zhang, Y.S., Guo, L.J., Chabchoub, A., He, J.S.: Higher-order rogue wave dynamics for derivative nonlinear Schrödinger equation. Rom. J. Phys. 62, 102 (2017)Google Scholar
  51. 51.
    Zhang, Y.S., Guo, L.J., Xu, S.W., Wu, Z.W., He, J.S.: The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 19, 1706–1722 (2014)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Wang, L., Zhu, Y.J., Wang, Z.Z., Qi, F.H., Guo, R.: Higher-order semirational solutions and nonlinear wave interactions for a derivative nonlinear Schrödingere quation. Commun. Nonlinear Sci. Numer. Simul. 33, 218–228 (2016)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Steudel, H.: The hierarchy of multi-soliton solutions of the derivative nonlinear Schrödinger equation. J. Phys. A Math. Gen. 36, 1931–1946 (2003)CrossRefzbMATHGoogle Scholar
  54. 54.
    Ichikawa, Y.H., Konno, K., Wadati, M., Sanuki, H.: Spiky soliton in circular polarized Alfv\(\acute{e}\)n wave. J. Phys. Soc. Jpn. 48, 279 (1980)CrossRefGoogle Scholar
  55. 55.
    Chen, X.J., Lam, W.K.: Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Phys. Rev. E 69, 066604 (2004)MathSciNetCrossRefGoogle Scholar
  56. 56.
    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)CrossRefGoogle Scholar
  57. 57.
    Chabchoub, A., Hoffmann, N., Onorato, M., Slunyaev, A., Sergeeva, A., Pelinovsky, E., Akhmediev, N.: Observation of a hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E 86, 056601 (2012)CrossRefGoogle Scholar
  58. 58.
    Wang, X., Yang, B., Chen, Y., Yang, Y.Q.: Higher-order localized waves in coupled nonlinear Schrödinger equations. Chin. Phys. Lett. 31, 090201 (2014)CrossRefGoogle Scholar
  59. 59.
    Wang, X., Li, Y.Q., Chen, Y.: Generalized Darboux transformation and localized waves in coupled Hirota equations. Wave Motion 51, 1149–1160 (2014)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Xu, T., Chen, Y., Lin, J.: Localized waves of the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics. Chin. Phys. B 26, 120201 (2017)CrossRefGoogle Scholar
  61. 61.
    Xu, T., Chen, Y.: Localized waves in three-component coupled nonlinear Schrödinger equation. Chin. Phys. B 25, 090201 (2016)CrossRefGoogle Scholar
  62. 62.
    Xu, T., Chen, Y.: Localised nonlinear waves in the three-component coupled Hirota equations. Z. Naturforsch. A 72, 1053–1070 (2017)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina
  2. 2.MOE International Joint Lab of Trustworthy SoftwareEast China Normal UniversityShanghaiChina
  3. 3.Department of PhysicsZhejiang Normal UniversityJinhuaChina

Personalised recommendations