Advertisement

Nonlinear Dynamics

, Volume 94, Issue 4, pp 3029–3040 | Cite as

Semi-rational solutions for a \((2+1)\)-dimensional Davey–Stewartson system on the surface water waves of finite depth

  • Yan Sun
  • Bo Tian
  • Yu-Qiang Yuan
  • Zhong Du
Original Paper
  • 77 Downloads

Abstract

Under investigation in this work is a \((2+1)\)-dimensional Davey–Stewartson system, which describes the surface water wave packets of finite depth. With respect to the velocity potential of the mean flow interacting with the surface wave and the amplitude of the surface wave packet, we derive two types of the solutions in terms of the Gramian, including the semi-rational solutions, and the solutions containing certain solitons and breathers based on the Kadomtsev–Petviashvili hierarchy reduction. Amplitude of the surface wave packet is graphically presented: (i) We find the interactions between the rogue waves/lump solitons and dark solitons, and the dark solitons keep their shapes unchanged after the interactions: The focusing/defocusing parameter does not affect the rogue wave and dark soliton, while the surface tension affects the locations of the rogue wave and dark soliton; (ii) We observe the interactions between the two dark–dark solitons, and three cases of the interactions between the dark solitons and breathers: The focusing/defocusing parameter only affects the propagation direction of the dark soliton, while the surface tension does not affect the two dark–dark solitons.

Keywords

Surface water waves of finite depth Semi-rational solutions \((2+1)\)-dimensional Davey–Stewartson system Kadomtsev–Petviashvili hierarchy reduction 

Notes

Acknowledgements

This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC:2017ZZ05), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. B 25, 16–43 (1983)CrossRefGoogle Scholar
  2. 2.
    Su, J.J., Gao, Y.T.: The Nth-order bright and dark solitons for the higher-order nonlinear Schrödinger equation in an optical fiber. Superlattice. Microstruct. 120, 697–719 (2018)CrossRefGoogle Scholar
  3. 3.
    Deng, G.F., Gao, Y.T.: Solitons for the (3+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equations in an optical fiber. Superlattice. Microstruct. 109, 345–359 (2017)CrossRefGoogle Scholar
  4. 4.
    Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A 468, 1716–1740 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gao, X.Y.: Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation. Appl. Math. Lett. 73, 143–149 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Clamond, D., Francius, M., Grue, J., Kharif, C.: Long time interaction of envelope solitons and freak wave formations. Eur. J. Mech. B Fluids 25, 536–553 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Liu, D.Y., Sun, W.R.: Rational solutions for the nonlocal sixth-order nonlinear Schrödinger equation. Appl. Math. Lett. 84, 63–69 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Porsezian, K., Nakkeeran, K.: Optical solitons in presence of Kerr dispersion and self-frequency shift. Phys. Rev. Lett. 76, 3955 (1996)CrossRefGoogle Scholar
  9. 9.
    Su, J.J., Gao, Y.T.: Integrability and solitons for the higher-order nonlinear Schrödinger equation with space-dependent coefficients in an optical fiber. Eur. Phys. J. Plus 133, 96 (2018)CrossRefGoogle Scholar
  10. 10.
    Belić, M., Petrović, N., Zhong, W.P., Xie, R.H., Chen, G.: Analytical light bullet solutions to the generalized \((3+1)\)-dimensional nonlinear Schrödinger equation. Phys. Rev. Lett. 101, 123904 (2008)CrossRefGoogle Scholar
  11. 11.
    Davey, A., Stewartson, K.: On three-dimensional packets of surface waves. Proc. R. Soc. A 338, 101–110 (1974)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zedan, H.A., Monaque, S.J.: The sine-cosine method for the Davey–Stewartson equations. Appl. Math. E 10, 103–111 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    McConnell, M., Fokas, A.S., Pelloni, B.: Localised coherent solutions of the DSI and DSII equations-a numerical study. Math. Comput. Simul. 69, 424–438 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zedan, H.A., Tantawy, S.S.: Solution of Davey–stewartson equations by homotopy perturbation method. Comput. Math. Math. Phys. 49, 1382–1388 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jafari, H., Sooraki, A., Talebi, Y., Biswas, A.: The first integral method and traveling wave solutions to Davey–Stewartson equation. Nonlinear Anal. 17, 182–193 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zhao, X.H., Tian, B., Xie, X.Y., Wu, X.Y., Sun, Y., Guo, Y.J.: Solitons, Bäcklund transformation and Lax pair for a \((2+1)\)-dimensional Davey–Stewartson system on surface waves of finite depth. Wave. Random. Complex 28, 356–366 (2018)CrossRefGoogle Scholar
  17. 17.
    Gupta, R.K., Kumar, V., Jiwari, R.: Exact and numerical solutions of coupled short pulse equation with time-dependent coefficients. Nonlinear Dyn. 79, 455–464 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kumar, D., Singh, J., Balednu, D.: A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Math. Method. Appl. Sci. 40, 5642–5653 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Singh, J., Kumar, D., Baleanu, D., Rathore, S.: An efficient numerical algorithm for the fractional Drinfeld–Sokolov-Wilson equation. Appl. Math. Comput. 335, 12–24 (2018)MathSciNetGoogle Scholar
  20. 20.
    Kumar, D., Agarwal, R.P., Singh, J.: A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. J. Comput. Appl. Math. 339, 405–413 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jiwari, R., Pandit, S., Mittal, R.C.: Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method. Comput. Phys. Commun. 183, 600–616 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kumar, D., Singh, J., Balednu, D.: Modified Kawahara equation within a fractional derivative with non-singular kernel. Therm. Sci. (2017).  https://doi.org/10.2298/TSCI160826008K CrossRefGoogle Scholar
  23. 23.
    Choi, J., Kumar, D., Singh, J., Swroop, R.: Analytical techniques for system of time fractional nonlinear differential equations. J. Korean Math. Soc. 54, 1209–1229 (2017)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Feng, Y.J., Gao, Y.T., Yu, X.: Soliton dynamics for a nonintegrable model of light-colloid interactive fluids. Nonlinear Dyn. 91, 29–38 (2018)CrossRefGoogle Scholar
  25. 25.
    Jia, T.T., Chai, Y.Z., Hao, H.Q.: Multi-soliton solutions and breathers for the generalized coupled nonlinear Hirota equations via the Hirota method. Superlattice. Microstruct. 105, 172–182 (2017)CrossRefGoogle Scholar
  26. 26.
    Huang, Q.M., Gao, Y.T.: Wronskian, Pfaffian and periodic wave solutions for a (2+1)-dimensional extended shallow water wave equation. Nonlinear Dyn. 89, 2855–2866 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Huang, Q.M., Gao, Y.T., Jia, S.L., Wang, Y.L., Deng, G.F.: Bilinear Bäcklund transformation, soliton and periodic wave solutions for a (3+1)-dimensional variable-coefficient generalized shallow water wave equation. Nonlinear Dyn. 87, 2529–2540 (2017)CrossRefGoogle Scholar
  28. 28.
    Xie, X.Y., Meng, G.Q.: Dark solitons for the (2+1)-dimensional Davey-Stewartson-like equations in the electrostatic wave packets. Nonlinear Dyn. 93, 779–783 (2018)CrossRefGoogle Scholar
  29. 29.
    Gao, X.Y.: Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid mechanics. Ocean Eng. 96, 245–247 (2015)CrossRefGoogle Scholar
  30. 30.
    Deng, G.F., Gao, Y.T.: Integrability, solitons, periodic and travelling waves of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles. Eur. Phys. J. Plus 132, 255 (2017)CrossRefGoogle Scholar
  31. 31.
    Jiwari, R., Kumar, V., Karan, R., Alshomrani, A.: Haar wavelet quasilinearization approach for MHD Falkner–Skan flow over permeable wall via Lie Group method. Int. J. Numer. Method. H. 27, 1332–1350 (2017)Google Scholar
  32. 32.
    Kumar, V., Guptab, R.K., Jiwarib, R.: Lie group analysis, numerical and non-traveling wave solutions for the \((2+1)\)-dimensional diffusion-advection equation with variable coefficients. Chin. Phys. B 23, 030201 (2014)CrossRefGoogle Scholar
  33. 33.
    Lan, Z.Z., Gao, B.: Lax pair, infinitely many conservation laws and solitons for a \((2+1)\)-dimensional Heisenberg ferromagnetic spin chain equation with time-dependent coefficients. Appl. Math. Lett. 79, 6–12 (2018)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Xie, X.Y., Meng, G.Q.: Collisions between the dark solitons for a nonlinear system in the geophysical fluid. Chaos, Solitons Fract. 107, 143–145 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ohta, Y., Yang, J.: Rogue waves in the Davey–Stewartson I equation. Phys. Rev. E 86, 036604 (2012)CrossRefGoogle Scholar
  36. 36.
    Chen, J.C., Chen, Y., Feng, B.F., Maruno, K.: Rational solutions to two-and one-dimensional multicomponent Yajima–Oikawa systems. Phys. Lett. A 379, 1510–1519 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Sun, Y., Tian, B., Liu, L., Chai, H.P., Yuan, Y.Q.: Rogue waves and lump solitons of the \((3+1)\)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves. Commun. Theor. Phys. 68, 693–700 (2017)CrossRefGoogle Scholar
  38. 38.
    Ohta, Y., Satsuma, J., Takahashi, D., Tokihiro, T.: An elementary introduction to sato theory. Prog. Theor. Phys. Supp. 94, 210–241 (1988)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Pelinovsky, E., Kharif, C.: Extreme Ocean Waves. Springer, Berlin (2008)CrossRefGoogle Scholar
  40. 40.
    Yin, J., Ai, H., Tian, L., Sun, M.: Energy rogue wave and its occurrence mechanism. Nonlinear Dyn. 82, 741–748 (2015)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Dysthe, K., Krogstad, H.E., Müller, P.: Oceanic rogue waves. Annu. Rev. Fluid Mech. 40, 287–310 (2008)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)CrossRefGoogle Scholar
  43. 43.
    Zhang, Y., Xu, Y.K., Shi, Y.B.: Rational solutions for a combined \((3+1)\)-dimensional generalized BKP equation. Nonlinear Dyn. 1, 1–11 (2018)zbMATHGoogle Scholar
  44. 44.
    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. 87, 67–73 (2017)CrossRefGoogle Scholar
  45. 45.
    Ankiewicz, A., Akhmediev, N.: Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions. Nonlinear Dyn. 91, 1931–1938 (2018)CrossRefGoogle Scholar
  46. 46.
    Kharif, C., Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B Fluids 22, 603–634 (2003)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Shats, M., Punzmann, H., Xia, H.: Capillary rogue waves. Phys. Rev. Lett. 104, 104503 (2010)CrossRefGoogle Scholar
  48. 48.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge Univeristy Press, Cambridge (2004)Google Scholar
  49. 49.
    Sato, M.: Soliton equations as dynamical systems on a infinite dimensional Grassmann manifolds (random systems and dynamical systems). RIMS Kokyuroku 439, 30 (1981)Google Scholar
  50. 50.
    Rao, J.G., Porsezian, K., He, J.S.: Semi-rational solutions of the third-type Davey–Stewartson equation. Chaos 27, 083115 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Information Photonics and Optical Communications, and School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina

Personalised recommendations