Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 685–699 | Cite as

Matter rogue waves and management by external potentials for coupled Gross–Pitaevskii equation

  • Fajun Yu
Original Paper


The analytical matter rogue wave solutions are reported for the coupled Gross–Pitaevskii equation by using the similarity transformation and Darboux transformation. We study the effect of the time-dependent linear and quadratic potentials (flying-bird potential) on the profiles and dynamics of non-autonomous rogue wave solution. A non-autonomous rogue wave and bright-dark rogue wave solutions are constructed and exhibited. The managements of external potential and the dynamic behaviors of the rogue wave solutions are investigated analytically. We present the general approach and use it to calculate non-autonomous rogue wave solutions and consider the potential applications for the rogue wave phenomena.


Non-autonomous rogue wave solution Darboux transformation Similarity transformation Gross–Pitaevskii equation 



This work was supported by the Natural Science Foundation of Liaoning Province, China (Grant No. 2013020056) and Project supported by the National Natural Science Foundation of China (Grant No.11301349).


  1. 1.
    Agrawal, G.P.: Nonlinear Fiber Optics. Academic, New York (1995)Google Scholar
  2. 2.
    Scott, A.C.: Launching a Davydov soliton: I. Soliton analysis. Phys. Scr. 29, 279–283 (1984)CrossRefGoogle Scholar
  3. 3.
    Bashkin, E.P., Vagov, A.V.: Instability and stratification of a two-component Bose-Einstein condensate. Phys. Rev. B 56, 6207–6212 (1997)CrossRefGoogle Scholar
  4. 4.
    Yan, Z.Y.: Financial rogue waves. Commun. Theor. Phys. 54, 947–949 (2010)CrossRefMATHGoogle Scholar
  5. 5.
    Dysthe, K., Krogstad, H.E., Müller, P.: Oceanic rogue waves. Annu. Rev. Fluid Mech. 40, 287–310 (2008)CrossRefGoogle Scholar
  6. 6.
    Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schr\(\ddot{o}\)dinger equation model. Phys. Rev. Lett. 85, 4502–4505 (2000)CrossRefGoogle Scholar
  7. 7.
    Kruglov, V.I., Peacock, A.C., Harvey, J.D.: Exact solutions of the generalized nonlinear Schr\(\ddot{o}\)dinger equation with distributed coefficients. Phys. Rev. E 71, 056619 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Tian, B., Shan, W.R., Zhang, C.Y., Wei, G.M., Gao, Y.T.: Schr\(\ddot{o}\)dinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation. Eur. Phys. J. B 47, 329–332 (2005)CrossRefGoogle Scholar
  9. 9.
    Agrawal, G.P.: Applications of Nonlinear Fiber Optics. Academic Press, New York (2001)Google Scholar
  10. 10.
    Zhang, J.L., Li, B.A., Wang, M.L.: The exact solutions and the relevant constraint conditions for two nonlinear Schr\(\ddot{o}\)dinger equations with variable coefficients. Chaos Solitons Fractals 39, 858–865 (2009)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    He, J.S., Li, Y.S.: Designable integrability of the variable coefficient nonlinear Schr\(\ddot{o}\)dinger equations. Stud. Appl. Math. 126, 1–15 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hao, R., Li, L., Li, Z., Xue, W., Zhou, G.: A new approach to exact soliton solutions and soliton interaction for the nonlinear Schr\(\ddot{o}\)dinger equation with variable coefficients. Opt. Commun. 236, 79–86 (2004)CrossRefGoogle Scholar
  13. 13.
    Guo, B.L., Ling, L.M.: Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schr\(\ddot{o}\)dinger equations. Chin. Phys. Lett. 28, 110202 (2011)CrossRefGoogle Scholar
  14. 14.
    Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schr\(\ddot{o}\)dinger equation. Phys. Rev. E 80, 026601 (2009)CrossRefGoogle Scholar
  15. 15.
    Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)CrossRefMATHGoogle Scholar
  16. 16.
    Sulem, C., Sulem, P.L.: The Nonlinear Schr\(\ddot{o}\)inger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999)Google Scholar
  17. 17.
    Chen, H.H., Liu, C.S.: Solitons in nonuniform media. Phys. Rev. Lett. 37, 693–697 (1976)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Ankiewicz, A.: Rogue Ocean Waves, URL of website (2009)
  19. 19.
    Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrodinger equation. Phys. Rev. E 80, 026601–026609 (2009)CrossRefGoogle Scholar
  20. 20.
    Ma, Y.C.: The perturbed plane-wave solution of the cubic Schr\(\ddot{o}\)dinger equation. Stud. Appl. Math. 60, 43–58 (1979)MathSciNetGoogle Scholar
  21. 21.
    Akhmediev, N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schr\(\ddot{o}\)dinger equation. Theor. Math. Phys. 69, 1089–1093 (1986)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Dysthe, K.B., Trulsen, K.: Note on breather type solutions of the NLS as models for freak-waves. Phys. Scr. T. 82, 48–52 (1999)CrossRefGoogle Scholar
  23. 23.
    Ten, I., Tomita, H.: Reports of RIAM Symposium No. 17SP1-2, Proceedings of a Symposium Held at Chikushi Campus, Kyushu University, Kasuga, Fukuoka, Japan (2006)Google Scholar
  24. 24.
    Voronovich, V.V., Shrira, V.I., Thomas, G.: Can bottom friction suppress ‘freak wave’ formation. J. Fluid Mech. 604, 263–296 (2008)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Akhmediev, N., Soto-Crespo, J.M., Ankiewicz, A.: Extreme waves that appear from nowhere: on the nature of rogue waves. Phys. Lett. A 373, 2137–2145 (2009)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Benjamin, T.B., Feir, J.E.: The disintegration of wavetrains in deep water. Part 1. J. Fluid Mech. 27, 417–431 (1967)CrossRefMATHGoogle Scholar
  27. 27.
    Bespalov, V.I., Talanov, V.I.: Filamentary structure of light beams in nonlinear liquids. JETP Lett. 3, 307–310 (1966)Google Scholar
  28. 28.
    Muller, P., Garrett, C., Osborne, A.: Rogue waves. Oceanography 18, 66–75 (2005)CrossRefGoogle Scholar
  29. 29.
    Chambarel, J., Kharif, C., Kimmoun, O.: Generation of two-dimensional steep water waves on finite depth with and without wind. Eur. J. Mech. B Fluids. 29, 132–142 (2010)CrossRefMATHGoogle Scholar
  30. 30.
    Kurkin, A.A., Pelinovsky, E.N.: Killer-Waves: Facts, Theory, and Modeling (Book in Russian). Nizhny Novgorod (2004)Google Scholar
  31. 31.
    Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)CrossRefGoogle Scholar
  32. 32.
    Akhmediev, N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman and Hall, London (1997)Google Scholar
  33. 33.
    Kivshar, Y.S., Agrawal, G.P.: Optical Solitons, From Fibers to Photonic Crystals. Academic, New York (2003)Google Scholar
  34. 34.
    Barnett, M.P., Capitani, J.F., Von Zur Gathen, J., Gerhard, J.: The Gauss-Bessel quadrature: a tool for the evaluation of Barnett–Coulson/Lowdin functions. Int. J. Quantum Chem. 100, 80–104 (2004)CrossRefGoogle Scholar
  35. 35.
    Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)Google Scholar
  36. 36.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York (1991)CrossRefMATHGoogle Scholar
  37. 37.
    Wadati, M.: Wave propagation in nonlinear lattice. I. J. Phys. Soc. Jpn. 38, 673–680 (1975)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Gao, Y.T., Tian, B.: Reply to: Comment on: Spherical Kadomtsev–Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation. Phys. Lett. A 361, 523–528 (2007)CrossRefMATHGoogle Scholar
  39. 39.
    Weiss, J., Tabor, M., Carnevale, G.: The Painleve property for partial differential equations. J. Math. Phys. 24, 522–529 (1983)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge Univ. Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  41. 41.
    Kharif, C., Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B/Fluids 22, 603–634 (2003)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Vector rogue waves in binary mixtures of Bose-Einstein condensates. Eur. Phys. J. Spec. Top. 185, 169–180 (2010)CrossRefGoogle Scholar
  43. 43.
    Odent, V., Taki, M., Louvergneaux, E.: Experimental spatial rogue patterns in an optical feedback system. Nat. Hazards Earth Syst. Sci. 10, 2727–2732 (2010)CrossRefGoogle Scholar
  44. 44.
    Yan, Z.Y., Konotop, V.V.: Exact solutions to three-dimensional generalized nonlinear Schr\(\ddot{o}\)dinger equations with varying potential and nonlinearities. Phys. Rev. E 80, 036607 (2009)CrossRefGoogle Scholar
  45. 45.
    Solli, D.R., Ropers, C., Jalali, B.: Active control of optical rogue waves for stimulated supercontinuum generation. Phys. Rev. Lett. 101, 233902 (2008)CrossRefGoogle Scholar
  46. 46.
    Fedele, F.: Rogue waves in oceanic turbulence. Phys. D 237, 2127–2131 (2008)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Shats, M., Punzmann, H., Xia, H.: Capillary rogue waves. Phys. Rev. Lett. 104, 104503 (2010)CrossRefGoogle Scholar
  48. 48.
    Ganshin, A.N., Efimov, V.B., Kolmakov, G.V., Mezhov-Deglin, L.P., McClintock, P.V.E.: Rogue events in the group velocity horizon. Phys. Rev. Lett. 101, 065303 (2008)CrossRefGoogle Scholar
  49. 49.
    Akhmediev, N., Pelinovsky, E.: Could rogue waves be used as efficient weapons against enemy ships. Eur. Phys. J. Spec. Top. Spec. Issue 185, 259–266 (2010)CrossRefGoogle Scholar
  50. 50.
    Yan, Z.Y.: Nonautonomous “rogons” in the inhomogeneous nonlinear Schr\(\ddot{o}\)dinger equation with variable coefficients. Phys. Lett. A 374, 672–679 (2010)CrossRefMATHGoogle Scholar
  51. 51.
    Dai, C.Q., Wang, X.G., Zhang, J.F.: Nonautonomous spatiotemporal localized structures in the inhomogeneous optical fibers: interaction and control. Ann. Phys. 326, 645–656 (2011)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Dai, C.Q., Zhou, G.Q., Zhang, J.F.: Controllable optical rogue waves in the femtosecond regime. Phys. Rev. E 85, 016603 (2012)CrossRefGoogle Scholar
  53. 53.
    Xu, S.W., He, J.S., Wang, L.H.: Two kinds of rogue waves of the general nonlinear Schrodinger equation with derivative. Eur. Phys. Lett. 97, 30007 (2012)CrossRefGoogle Scholar
  54. 54.
    Zhao, L.C., Liu, J.: Localized nonlinear waves in a two-mode nonlinear fiber. J. Opt. Soc. Am. B 29, 3119–3127 (2012)CrossRefGoogle Scholar
  55. 55.
    Wen, L., et al.: Matter rogue wave in Bose-Einstein condensates with attractive atomic interaction. Eur. Phys. J. D 64, 473–478 (2011)Google Scholar
  56. 56.
    Yang, G.Y., et al.: Peregrine rogue waves induced by the interaction between a continuous wave and a soliton. Phys. Rev. E 85, 046608 (2012)CrossRefGoogle Scholar
  57. 57.
    Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)CrossRefGoogle Scholar
  58. 58.
    Wang, Y.Y., Dai, C.Q., Wang, X.G.: Stable localized spatial solitons in PT-symmetric potentials with power-law nonlinearity. Nonlinear Dyn. 77, 1323–1330 (2014)CrossRefMathSciNetGoogle Scholar
  59. 59.
    Dai, C.Q., Zhu, H.P.: Superposed Akhmediev breather of the 3-dimensional generalized nonlinear Schrödinger equation with external potentials. Ann. Phys. 341, 142–152 (2014)CrossRefMathSciNetGoogle Scholar
  60. 60.
    Zhu, H.P.: Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides. Nonlinear Dyn. 72, 873–882 (2013)CrossRefGoogle Scholar
  61. 61.
    Wu, X.F., Hua, G.S., Ma, Z.Y.: Novel rogue waves in an inhomogenous nonlinear medium with external potentials. Commun. Nonlinear Sci. Numer. Simul. 18, 3325–3336 (2013)CrossRefMathSciNetGoogle Scholar
  62. 62.
    Zhu, H.P.: Spatiotemporal solitons on cnoidal wave backgrounds in three media with different distributed transverse diffraction and dispersion. Nonlinear Dyn. 76, 1651–1659 (2014)CrossRefGoogle Scholar
  63. 63.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)CrossRefGoogle Scholar
  64. 64.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons. J. Mod. Opt. 57, 1456–1472 (2010)CrossRefMATHGoogle Scholar
  65. 65.
    Mani Rajan, M.S., Mahalingam, A.: Multi-soliton propagation in a generalized inhomogeneous nonlinear Schrödinger-Maxwell-Bloch system with loss/gain driven by an external potential. J. Math. Phys. 54, 043514 (2013)CrossRefMathSciNetGoogle Scholar
  66. 66.
    Mani Rajan, M.S., Mahalingam, A., Uthayakumar, A.: Nonlinear tunneling of nonautonomous optical solitons in combined nonlinear Schrödinger and Maxwell-Bloch systems. J. Opt. 14, 105204 (2012)CrossRefGoogle Scholar
  67. 67.
    Zhao, L.C., He, S.L.: Matter wave solitons in coupled system with external potentials. Phys. Lett. A 375, 3017–3020 (2011)CrossRefGoogle Scholar
  68. 68.
    Zhao, L.C., Yang, Z.Y., Ling, L.M., Liu, J.: Precisely controllable bright nonautonomous solitons in Bose-Einstein condensate. Phys. Lett. A 375, 1839–1842 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematics and Systematic SciencesShenyang Normal UniversityShenyangChina

Personalised recommendations