Nonlinear Dynamics

, Volume 93, Issue 2, pp 585–597 | Cite as

Higher-order rogue wave solutions of a general coupled nonlinear Fokas–Lenells system

  • Jianwen Yang
  • Yi Zhang
Original Paper


In this paper, we construct a generalized Darboux transformation (DT) of the general coupled nonlinear Fokas–Lenells system. And using this generalized DT, new exact higher-order rogue wave solutions of the system are obtained. In addition, the dynamics of the solutions are discussed and shown to exhibit interesting structure.


The general coupled nonlinear Fokas–Lenells system Generalized Darboux transformation Rogue wave 



The authors would like to thank Professor J. S. He and Dr. S. W. Xu for their enthusiastic help.


  1. 1.
    Mller, P., Garrett, Ch., Osborne, A.: The fourteenth Aha Hulikoa Hawaiian winter workshop. Oceanography 18, 66–75 (2005)CrossRefGoogle Scholar
  2. 2.
    Wang, Y.Y., Dai, C.Q., Zhou, G.Q., Fan, Y., Chen, L.: Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium. Nonlinear Dyn. 3, 1–7 (1993)Google Scholar
  3. 3.
    Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Hasegawa, A., Brinkman, W.F.: Tunable coherent ir and fir sources utilizing modulational instability. IEEE J. Quantum Electron. 16, 694–697 (1980)CrossRefGoogle Scholar
  5. 5.
    Anderson, D., Lisak, M.: Modulational instability of coherent optical-fiber transmission signals. Opt. Lett. 9, 468–470 (1984)CrossRefGoogle Scholar
  6. 6.
    Akhmediev, N.N., Korneev, V.I., Mitskevich, N.V.: Modulation instability of a continuous signal in an optical fiber taking into account third-order dispersion. Radiophys. Quantum Electron. 33, 95–100 (1990)CrossRefGoogle Scholar
  7. 7.
    Tai, K., Tomita, J.L., Jewell, J.L., Hasegawa, A.: Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability. Appl. Phys. Lett. 49, 236–238 (1986)CrossRefGoogle Scholar
  8. 8.
    Garrett, C., Gemmrich, J.: Rogue waves. Phys. Today 62, 62–63 (2009)CrossRefGoogle Scholar
  9. 9.
    Onorato, M., Waseda, T., Toffoli, A., Cavaleri, L., Gramstad, O., Janssen, P.A.E.M., Kinoshita, T., Monbaliu, J., Mori, N., Osborne, A.R., Serio, M., Stansberg, C.T., Tamura, H., Trulsen, K.: Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Phys. Rev. Lett. 102, 114502 (2009)CrossRefGoogle Scholar
  10. 10.
    Akhmediev, N.N., Korneev, V.I., Mitskevich, N.V.: N-modulation signals in a single-mode optical fiber with allowance for nonlinearity. Zh. Eksp. Teor. Fiz. 94, 159–170 (1988)Google Scholar
  11. 11.
    Chow, K.W., Chan, H.N., Kedziora, D.J., Kedziora, D.J., Grimshaw, R.H.J.: Rogue wave modes for the long wave-short wave resonance model. J. Phys. Soc. Jpn. 82(7), 4001 (2013)Google Scholar
  12. 12.
    Wu, C.F., Grimshaw, R.H.J., Chow, K.W., Chan, H.N.: A coupled ‘AB’ system: rogue waves and modulation instabilities. Chaos 25, 103113 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equation and their solutions. Anziam J. 25(1), 16–43 (1983)zbMATHGoogle Scholar
  14. 14.
    Ankiewicz, A., Kedziora, D.J., Akhmediev, N.: Rogue wave triplets. Phys. Lett. A. 375, 2782–2785 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    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 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ohta, Y., Yang, J.K.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468, 1716–1740 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zhang, H.Q., Ma, W.X.: Lump solutions to the (2+1)-dimensional sawada-kotera equation. Nonlinear Dyn. 87(4), 1–6 (2016)MathSciNetGoogle Scholar
  19. 19.
    Zhang, Y., Sun, Y.B., Xiang, W.: The rogue waves of the KP equation with self-consistent sources. Appl. Math. Comput. 263, 204–213 (2015)MathSciNetGoogle Scholar
  20. 20.
    Lenells, J.: Exactly solvable model for nonlinear pulse propagation in optical fibers. Stud. Appl. Math. 123, 215–232 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhao, P., Fan, E., Hou, Y.: Algebro-geometric solutions and their reductions for the Fokas–Lenells hierarchy. J. Nonlinear Math. Phys. 20, 355–393 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chen, S., Song, L.Y.: Peregrine solitons and algebraic soliton pairs in Kerr media considering space-time correction. Phys. Lett. A 378, 1228–1232 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xu, S., He, J.S., Cheng, Y., Porseizan, K.: The n-th order rogue waves of Fokas–Lenells equation. Math. Methods Appl. Sci. 38, 1106–1126 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, L., Geng, C., Zhang, L.L., Zhao, Y.C.: Characteristics of the nonautonomous breathers and rogue waves in a generalized Lenells-Fokas equation. EPL 108(5), 50009 (2014)CrossRefGoogle Scholar
  25. 25.
    Matsuno, Y.: A direct method of solution for the Fokas–Lenells derivative nonlinear Schrödinger equation: I. Bright soliton solutions. J. Phys. A Math. Theor. 45, 235202 (2012)CrossRefzbMATHGoogle Scholar
  26. 26.
    Matsuno, Y.: A direct method of solution for the Fokas–Lenells derivative nonlinear Schrödinger equation: II. Dark soliton solutions. J. Phys. A Math. Theor. 45, 475202 (2012)CrossRefzbMATHGoogle Scholar
  27. 27.
    Ma, W.X., Ding, Q., Zhang, W.G., Lu, B.Q.: Binary non-linearization of Lax pairs of Kaup–Newell soliton hierarchy. Nuov. Cim. B. 111, 1135–1149 (1996)CrossRefGoogle Scholar
  28. 28.
    Chen, X.J., Lan, W.K.: Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Phys. Rev. E 69, 066604 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lenells, J.: The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line. Physica D Nonlinear Phenom. 240(6), 512–525 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Guo, B., Ling, L.M., Liu, Q.P.: High-order solutions and generalized darboux transformations of derivative nonlinear Schrödinger equations. Stud. Appl. Math. 130, 317–344 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wen, X.Y., Yang, Y., Yan, Z.: Generalized perturbation (n, M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation. Phys. Rev. E 92, 012917 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lü, X., Peng, M.: Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics. Commun. Nonlinear Sci. Numer. Simul. 18, 2304–2312 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lü, X.: Madeling fluid description on a generalized mixed nonlinear Schrödinger equation. Nonlinear Dyn. 81, 239–247 (2015)CrossRefzbMATHGoogle Scholar
  34. 34.
    Tsuchida, T.: New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems. J. Math. Phys. 51, 053511 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tsuchida, T., Wadati, M.: New integrable systems of derivative nonlinear Schrödinger equations with multiple components. Phys. Lett. A 257, 53–64 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Fordy, A.P.: Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces. J. Phys. A 17, 1235–1246 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Guo, B.L., Ling, L.M.: Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation. J. Math. Phys. 53, 073506 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Morris, H.C., Dodd, R.K.: The two component derivative nonlinear Schrödinger equation. Phys. Scr. 20, 505–508 (1979)CrossRefzbMATHGoogle Scholar
  39. 39.
    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
  40. 40.
    Manakov, S.V.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP 38, 248–253 (1974)Google Scholar
  41. 41.
    Zhang, Y., Yang, J.W., Chow, K.W., Wu, C.F.: Solitons, breathers and rogue waves for the coupled Fokas–Lenells system via Darboux transformation. Nonlin. Anal. Real World Appl. 33, 237–252 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    He, J.S., Zhang, H.R., Wang, L.H., Porsezian, K., Fokas, A.S.: Generating mechanism for higher-order rogue waves. Phys. Rev. E 87, 052914 (2013)CrossRefGoogle Scholar
  43. 43.
    Xu, S.W., He, J.S.: The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A 44, 305203 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China

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