Nonlinear Dynamics

, Volume 93, Issue 4, pp 2169–2184 | Cite as

General high-order rogue waves to nonlinear Schrödinger–Boussinesq equation with the dynamical analysis

  • Xiaoen Zhang
  • Yong ChenEmail author
Original Paper


General high-order rogue waves of the nonlinear Schrödinger–Boussinesq equation are obtained by the KP-hierarchy reduction theory, and the N-order rogue waves are expressed with the determinants, whose entries are all algebraic forms, which is shown in the theorem. It is found that the fundamental first-order rogue waves can be classified into three patterns: four-petal state, dark state, bright state by choosing different values of parameter \(\alpha \). An interesting phenomenon is discovered as the evolution of the parameter \(\alpha \): the rogue wave changes from four-petal state to dark state, whereafter bright state, which are consistent with the change in the corresponding critical points to the function of two variables. Furthermore, the dynamical property of second-order and third-order rogue waves is plotted, which can be regarded as the nonlinear superposition of the fundamental first-order rogue waves.


High-order rogue waves Nonlinear Schrödinger–Boussinesq equation KP-hierarchy reduction technique 



We would like to express our sincere thanks to SY Lou, WX Ma, EG Fan, ZY Yan, XY Tang, JC Chen, X Wang and other members of our discussion group for their valuable comments.

Compliance with ethical standards

Conflict of interest

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


  1. 1.
    Onorato, M., Osborne, A.R., Srio, M.: Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys. Rev. Lett. 96, 014503 (2006)CrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Peterson, P., Soomere, T., Engelbrecht, J., Groesen, E.V.: Soliton interaction as a possible model for extreme waves in shallow water. Nonlinear Process. Geophys. 10, 503–510 (2003)CrossRefGoogle Scholar
  4. 4.
    Pelinovsky, E., Kharif, C., Talipova, T.: Large-amplitude long wave interaction with a vertical wall. Eur. J. Mech. B Fluid 27, 409–418 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1058 (2007)CrossRefGoogle Scholar
  6. 6.
    Pierangeli, D., Mei, F.D., Conti, C., Agranat, A.J., DelRe, E.: Spatial rogue waves in photorefractive ferroelectrics. Phys. Rev. Lett. 115, 093901 (2015)CrossRefGoogle Scholar
  7. 7.
    Akhmediev, N., Dudley, J.M., Solli, D.R., Turitsyn, S.K.: Recent progress in investigating optical rogue waves. J. Opt. 15, 060201 (2013)CrossRefGoogle Scholar
  8. 8.
    Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80, 033610 (2009)CrossRefGoogle Scholar
  9. 9.
    Moslem, W.M.: Langmuir rogue wave in electron–positron plasmas. Phys. Plasmas 18, 032301 (2011)CrossRefGoogle Scholar
  10. 10.
    Yan, Z.Y.: Vector financial rogue waves. Phys. Lett. A 375, 4274–4279 (2011)CrossRefzbMATHGoogle Scholar
  11. 11.
    Ohta, Y., Yang, J.K.: Genera high-order rogue wvae and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. Lond. Sect. A 468, 1716–1740 (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, 026601 (2009)CrossRefGoogle Scholar
  13. 13.
    Guo, B.L., Ling, L.M., Liu, Q.P.: High-order solution and generalized Darboux transformation of derivative nonlinear Schrödinger equation. Stud. Appl. Math. 130, 317–344 (2012)CrossRefGoogle Scholar
  14. 14.
    Ling, L.M., Guo, B.L., Zhao, L.C.: High-order rogue waves in vector nonlinear Schrödinger equation. Phys. Rev. E 89, 041201 (2014)CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Wang, Y.Y., Liang, C., Dai, C.Q., Zheng, J., Fan, Y.: Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity. Nonlinear Dyn. 90, 1269–1275 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dai, C.Q., Zhou, G.Q., Chen, R.P., Lai, X.J., Zheng, J.: Vector multipole and vortex solitons in two-dimensional Kerr media. Nonlinear Dyn. 88, 2629–2635 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dai, C.Q., Wang, Y.Y., Fan, Y., Yu, D.G.: Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under PT-symmetric potentials. Nonlinear Dyn. 92, 1351–1358 (2018)CrossRefGoogle Scholar
  19. 19.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. B 25, 16–43 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010)CrossRefGoogle Scholar
  21. 21.
    Zhao, L.C., Xin, G.G., Yang, Z.Y.: Rogue-wave pattern transition induced by relative frequency. Phys. Rev. E 90, 022918 (2014)CrossRefGoogle Scholar
  22. 22.
    Chabchoub, A., Hoffmann, N., Onorato, M., Slunyaev, A., Sergeeva, A., Pelinovsky, E., Akhmediev, N.: Observation of hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E 86, 056601 (2012)CrossRefGoogle Scholar
  23. 23.
    Guo, B.L., Ling, L.M.: Rogue wave, breathers and bright–dark-rogue solutions for the coupled Schrödinger equations. Chin. Phys. Lett. 28, 110202 (2011)CrossRefGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xu, T., Chen, Y., Lin, J.: Localized waves of the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics. Chin. Phys. B 26, 120200 (2017)Google Scholar
  26. 26.
    Wei, J., Wang, X., Geng, X.G.: Periodic and rational solutions of the reduced Maxwell–Bloch equations. Commun. Nonlinear Sci. Numer. Simul. 59, 1–14 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Liu, Y.K., Li, B., An, H.L.: General high-order breathers, lumps in the (\(2+1\))-dimensional Boussinesq equation. Nonlinear Dyn. (2018). Google Scholar
  28. 28.
    Zhang, X.E., Chen, Y., Tang, X.Y.: Rogue wave and a pair of resonance stripe solitons to a reduced generalized (\(3+1\))-dimensional KP equation. arXiv:1610.09507
  29. 29.
    Zhang, X.E., Chen, Y.: Rogue wave and a pair of resonance stripe solitons to a reduced (\(3+1\))-dimensional Jimbo–Miwa equation. Commun. Nonlinear Sci. Numer. Simul. 52, 24–31 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhang, X.E., Chen, Y.: Deformation rogue wave to the (\(2+1\))-dimensional KdV equation. Nonlinear Dyn. 90, 755–763 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS Kyoto Univ. 19, 943–1001 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ohta, Y.: Wronskian solutions of soliton equations. RIMS kôkyûroku 684, 1–17 (1989)MathSciNetGoogle Scholar
  33. 33.
    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
  34. 34.
    Feng, B.F.: General \(N\)-soliton solution to a vector nonlinear Schrödinger equation. J. Phys. A Math. Theor. 47, 355203 (2014)CrossRefzbMATHGoogle Scholar
  35. 35.
    Ling, L.M., Zhao, L.C., Guo, B.L.: Darboux transformation and multi-dark soliton for \(N\)-component nonlinear Schrödinger equations. Nonlinearity 28, 3243–3261 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Chen, J.C., Chen, Y., Feng, B.F., Maruno, K.I., Ohta, Y.: An integrable semi-discretization of the coupled Yajima–Oikawa system. J. Phys. A Math. Theor. 49, 165201 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Chen, J.C., Chen, Y., Feng, B.F., Maruno, K.I., Ohta, Y.: General high-order rogue waves of the (\(1+1\))-dimensional Yajima–Oikawa system. arXiv:1709.03781
  38. 38.
    Chen, J.C., Chen, Y., Feng, B.F., Maruno, K.I.: Rational solutions to two-and one-dimensional multicomponent Yajima–Oikawa systems. Phys. Lett. A 379, 1510–1519 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Chen, J.C., Feng, B.F., Chen, Y., Ma, Z.Y.: General bright–dark soliton solutions to (\(2+1\))-dimensional multi-component long-wave–short-wave resonance interaction system. Nonlinear Dyn. 88, 1–16 (2017)CrossRefzbMATHGoogle Scholar
  40. 40.
    Han, Z., Chen, Y., Chen, J.C.: General \(N\)-dark soliton solutions of the multi-component Mel’nikov system. J. Phys. Soc. Jpn. 86, 074005 (2017)CrossRefGoogle Scholar
  41. 41.
    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)CrossRefGoogle Scholar
  42. 42.
    Sun, B.N., Wazwaz, A.M.: Interaction of lumps and dark solitons in the Mel’nikov equation. Nonlinear Dyn. (2018). Google Scholar
  43. 43.
    Wazwaz, A.M.: Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations. Nonlinear Dyn. 85, 731–737 (2016)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wazwaz, A.M., El-Tantawy, S.A.: Solving the (\(3+1\))-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn. 88, 3017–3021 (2017)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Rao, N.N.: Exact solutions of coupled scalar field equations. J. Phys. A Math. Gen. 22, 4813–4825 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Singh, S.V., Rao, N.N., Shukla, P.K.: Nonlinearly coupled Langmuir and dust-acoustic waves in a dusty plasma. J. Plasma Phys. 3, 551–567 (1998)CrossRefGoogle Scholar
  47. 47.
    Hase, Y., Satsuma, J.: An \(N\)-soliton solutions for the nonlinear Schrödinger equation coupled to the Boussinesq equation. J. Phys. Soc. Jpn. 57, 679–682 (1988)CrossRefGoogle Scholar
  48. 48.
    Mu, G., Qin, Z.Y.: Rogue waves for the coupled Schrödinger–Boussinesq equation and the coupled higgs equation. J. Phys. Soc. Jpn. 81, 084001 (2012)CrossRefGoogle Scholar
  49. 49.
    Lu, C.N., Fu, C., Yang, H.W.: Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions. Appl. Math. Comput. 327, 104–116 (2018)MathSciNetGoogle Scholar
  50. 50.
    Xu, X.X.: An integrable coupling hierarchy of the Mkdv-integrable systems, its hamiltonian structure and corresponding nonisospectral integrable hierarchy. Appl. Math. Comput. 216, 344–353 (2010)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Xu, T., Chen, Y.: Darboux transformation of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization. Commun. Nonlinear Sci. Numer. Simul. 57, 276–289 (2018)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Tang, L.Y., Fan, J.C.: A family of liouville integrable lattice equations and its conservation laws. Appl. Math. Comput. 217, 1907–1912 (2010)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Li, X.Y., Li, Y.X., Yang, H.X.: Two families of liouville integrable lattice equations. Appl. Math. Comput. 217, 8671–8682 (2011)MathSciNetzbMATHGoogle 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

Personalised recommendations