Darboux transformation of a new generalized nonlinear Schrödinger equation: soliton solutions, breather solutions, and rogue wave solutions

Original Paper
  • 54 Downloads

Abstract

In this paper, a new generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. Firstly, the n-fold Darboux transformation (DT) of the GNLS equation is constructed. Then, the soliton solutions, breather solutions, and rogue wave solutions of the GNLS equation are studied based on the DT by choosing different seed solutions. Furthermore, the dynamic features of these solutions are explicitly delineated through some figures with the help of Maple software.

Keywords

Generalized nonlinear Schrödinger equation Darboux transformation Soliton solutions Breather solutions Rogue wave solutions 

Notes

Acknowledgements

Thank our partners for their help. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Especially, we are very grateful to the editor and reviewers for their constructive comments and suggestions. This research has been supported by the Natural Science Basic Research Program of Shaanxi (No. 2017JM1024).

References

  1. 1.
    Johnson, R.S.: On the modulation of water waves in the neighbourhood of \(kh\) \(\approx \) 1.363. Proc. R. Soc. A 357(1689), 131–141 (1977)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Henderson, K.L., Peregrine, D.H., Dold, J.W.: Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29(4), 341–361 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dudley, J.M., Genty, G., Dias, F., Kibler, B., Akhmediev, N.: Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation. Opt. Express 17(24), 21497–21508 (2009)CrossRefGoogle Scholar
  4. 4.
    Fokas, A.S.: On a class of physically important integrable equations. Physica D 87(1–4), 145–150 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Lenells, J.: Dressing for a novel integrable generalization of the nonlinear Schrödinger equation. J. Nonlinear Sci. 20(6), 709–722 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Matveev, V.B., Salle, M.A.: Darboux transformation and solitons. J. Neurochem. 42(6), 1667–1676 (1991)MATHGoogle Scholar
  7. 7.
    Xu, S., He, J., Wang, L.: The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A Math. Theor. 44(30), 6629–6636 (2011)CrossRefGoogle Scholar
  8. 8.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  9. 9.
    Tanolu, G.: Hirota method for solving reaction–diffusion equations with generalized nonlinearity. Int. J. Nonlinear Sci. 1(1), 1479–3889 (2006)MathSciNetGoogle Scholar
  10. 10.
    Kakei, S., Sasa, N., Satsuma, J.: Bilinearization of a generalized derivative nonlinear Schrödinger equation. J. Phys. Soc. Jpn. 64(5), 1519–1523 (2012)CrossRefMATHGoogle Scholar
  11. 11.
    Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373(6), 675–678 (2009)CrossRefMATHGoogle Scholar
  12. 12.
    Tai, K., Hasegawa, A., Tomita, A.: Observation of modulational instability in optical fibers. Phys. Rev. Lett. 56(2), 135 (1986)CrossRefGoogle Scholar
  13. 13.
    Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450(7172), 1054–7 (2007)CrossRefGoogle Scholar
  14. 14.
    Wabnitz, S., Finot, C., Fatome, J., Millot, G.: Shallow water rogue wavetrains in nonlinear optical fibers. Phys. Lett. A 377(12), 932–939 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ganshin, A.N., Efimov, V.B., Kolmakov, G.V., Mezhov-Deglin, L.P., Mcclintock, P.V.: Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. Phys. Rev. Lett. 101(6), 065–303 (2008)CrossRefMATHGoogle Scholar
  16. 16.
    Mlejnek, M., Wright, E.M., Moloney, J.V.: Femtosecond pulse propagation in argon: a pressure dependence study. Phys. Rev. E: Stat. Phys. Plasmas Fluids 58(4), 4903–4910 (1998)CrossRefGoogle Scholar
  17. 17.
    Akhmediev, N.N., Korneev, V.I., Mitskevich, N.V.: N-modulation signals in a single-mode optical fiber with allowance for nonlinearity. Zhurnal Eksperimentalnoi I Teroreticheskoi Fiziki 94(1), 159–170 (1988)Google Scholar
  18. 18.
    Efimov, V.B., Ganshin, A.N., Kolmakov, G.V., Mcclintock, P.V.E., Mezhov-Deglin, L.P.: Rogue waves in superfluid helium. Eur. Phys. J. Spec. Top. 185(1), 181–193 (2010)CrossRefMATHGoogle Scholar
  19. 19.
    Stenflo, L., Marklund, M.: Rogue waves in the atmosphere. J. Plasma Phys. 76(3–4), 293–295 (2010)CrossRefGoogle Scholar
  20. 20.
    Yan, Z.: Financial rogue waves appearing in the coupled nonlinear volatility and option pricing model. Soc. Politics 18(3), 441–468 (2011)CrossRefGoogle Scholar
  21. 21.
    Akhmediev, N.N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69(2), 1089–1093 (1986)CrossRefMATHGoogle Scholar
  22. 22.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equation and their solutions. J. Aust. Math. Soc. 25(1), 16–43 (1983)CrossRefMATHGoogle Scholar
  23. 23.
    Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(2), 026601 (2009)CrossRefMATHGoogle Scholar
  24. 24.
    Ling, L., Guo, B., Zhao, L.C.: High-order rogue waves in vector nonlinear Schrödinger equations. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 89(4), 041201 (2014)CrossRefGoogle Scholar
  25. 25.
    Ankiewicz, A., Soto-Crespo, J.M., Akhmediev, N.: Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 81(4 Pt 2), 046602 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rao, J.G., Liu, Y.B., Qian, C., He, J.S.: Rogue waves and hybrid solutions of the Boussinesq equation. Z. Fr. Naturforschung A 72(4), 026601 (2017)Google Scholar
  27. 27.
    Akhmediev, N., Soto-Crespo, J.M., Devine, N., Hoffmann, N.P.: Rogue wave spectra of the Sasasatsuma equation. Physica D 294, 37–42 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    He, J., Xu, S., Porsezian, K.: Rogue waves of the Fokas–Lenells equation. J. Phys. Soc. Jpn. 81(12), 4007 (2012)Google Scholar
  29. 29.
    Graeff, C.F.O., Stutzmann, M., Brandt, M.S.: Akhmediev breathers, Ma solitons, and general breathers from rogue waves: a case study in the Manakov system. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 88(2), 022918 (2013)CrossRefGoogle Scholar
  30. 30.
    Chen, S., Song, L.: Rogue waves in coupled Hirota systems. Phys. Rev. E Stat. Phys. Plasmas, Fluids 87(87), 83–99 (2013)Google Scholar
  31. 31.
    Wang, X., Liu, C., Wang, L.: Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations. J. Math. Anal. Appl. 449(2), 1534–1552 (2017)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zuo, D.W., Gao, Y.T., Feng, Y.J., Xue, L.: Rogue-wave interaction for a higher-order nonlinear Schrödinger–Maxwell–Bloch system in the optical-fiber communication. Nonlinear Dyn. 78(4), 2309–2318 (2014)CrossRefMATHGoogle Scholar
  33. 33.
    Chen, J., Chen, Y., Feng, B.F., Maruno, K.: Rational solutions to multicomponent Yajima–Oikawa systems: from two dimension to one dimension. Physics 40(Suppl 4), 737–756 (2014)Google Scholar
  34. 34.
    Li, L., Wu, Z., Wang, L., He, J.: High-order rogue waves for the Hirota equation. Ann. Phys. 334(7), 198–211 (2013)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zhang, Y., Li, C., He, J.: Rogue waves in a resonant erbium-doped fiber system with higher-order effects. Appl. Math. Comput. 273, 826–841 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Geng, X., Lv, Y.: Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation. Nonlinear Dyn. 69(4), 1621–1630 (2012)CrossRefMATHGoogle Scholar
  37. 37.
    Gu, C., Hu, H., Zhou, Z.: Darboux Transformations in Integrable Systems. Springer, Berlin (2004)Google Scholar
  38. 38.
    Neamaty, A., Mosazadeh, S., Majidi, A.: Generalized darboux transformation and nth order rogue wave solution of a general coupled nonlinear Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 20(2), 401–420 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Wang, D.S., Chen, F., Wen, X.Y.: Darboux transformation of the general Hirota equation: multisoliton solutions, breather solutions, and rogue wave solutions. Adv. Differ. Equ. 2016(1), 67 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zhang, Y., Yang, J.W., Chow, K.W., Wu, C.F.: Solitons, breathers and rogue waves for the coupled Fokaslenells system via Darboux transformation. Nonlinear Anal. Real World Appl. 33, 237–252 (2017)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zhaqilao, : On nth-order rogue wave solution to the generalized nonlinear Schrödinger equation. Phys. Lett. A 377(12), 855–859 (2013)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Shan, S., Li, C., He, J.: On rogue wave in the Kundu-DNLS equation. Commun. Nonlinear Sci. Numer. Simul. 18(12), 3337–3349 (2013)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Wen, L.L., Zhang, H.Q.: Rogue wave solutions of the \((2+1)\)-dimensional derivative nonlinear Schrödinger equation. Nonlinear Dyn. 86(2), 877–889 (2016)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Guo, R., Zhao, H.H., Wang, Y.: A higher-order coupled nonlinear Schrödinger system: solitons, breathers, and rogue wave solutions. Nonlinear Dyn. 83(4), 2475–2484 (2016)CrossRefMATHGoogle Scholar
  45. 45.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 85(2), 066601 (2012)CrossRefGoogle Scholar
  46. 46.
    Qiu, D., Zhang, Y., He, J.: The rogue wave solutions of a new \((2+1)\)-dimensional equation. Commun. Nonlinear Sci. Numer. Simul. 30, 307–315 (2015)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Lenells, J.: Exactly solvable model for nonlinear pulse propagation in optical fibers. Stud. Appl. Math. 123(2), 215–232 (2009)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Xing, L., Tian, B.: Novel behavior and properties for the nonlinear pulse propagation in optical fibers. EPL 97(1), 10005 (2012)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

Personalised recommendations