Skip to main content
SpringerLink
Log in

Dynamic behaviors of the breather solutions for the AB system in fluid mechanics

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Under investigation in this paper is the AB system, which describes marginally unstable baroclinic wave packets in geophysical fluids. Through symbolic computation, Lax pair and conservation laws are derived and the Darboux transformation is constructed for this system. Furthermore, three types of breathers on the continuous wave (cw) background are generated via the obtained Darboux transformation. The following contents are mainly discussed by figures plotted: (1) Modulation instability processes of the Akhmediev breathers in the presence of small perturbations; (2) Propagations characteristics of Ma solitons; (3) Dynamic features of the breathers evolving periodically along the straight line with a certain angle of z-axis and t-axis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Ji, J.: The double Wronskian solutions of a non-isospectral Kadomtsev–Petviashvili equation. Phys. Lett. A 372(39), 6074–6081 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Sun, F.W., Cai, J.X., Gao, Y.T.: Analytic localized solitonic excitations for the (2+1)-dimensional variable-coefficient breaking soliton model in fluids and plasmas. Nonlinear Dyn. 70, 1889–1901 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Tian, S.F., Zhang, T.T., Zhang, H.Q.: Darboux transformation and new periodic wave solutions of generalized derivative nonlinear Schröinger equation. Phys. Scr. 80, 065013 (2009)

    Article  Google Scholar 

  4. Kohl, R., Biswas, A., Milovic, D., Zerrad, E.: Optical soliton perturbation in a non-Kerr law media. Opt. Laser Technol. 40, 647–655 (2008)

    Article  Google Scholar 

  5. Shidfar, A., Molabahrami, A., Babaei, A., Yazdanian, A.: A series solution of the Cauchy problem for the generalized d-dimensional Schrödinger equation with a power-law nonlinearity. Comput. Math. Appl. 59, 1500–1508 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo, R., Tian, B.: Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation. Commun. Nonlinear Sci. Numer. Simul. 17, 3189–3203 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ghodrat, E., Anjan, B.: The G′/G method and 1-soliton solution of Davey–Stewartson equation. Math. Comput. Model. 53(5–6), 694–698 (2011)

    MATH  Google Scholar 

  8. Ghodrat, E., Krishnan, E.V., Manel, L., Essaid, Z., Anjan, B.: Analytical and numerical solutions for Davey–Stewartson equation with power law nonlinearity. Waves Random Complex Media 21(4), 559–590 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hossein, J., Atefe, S., Yahya, T., Anjan, B.: The first integral method and traveling wave solutions to Davey–Stewartson equation. Nonlinear Anal. Mod. Cont. 17(2), 182–193 (2012)

    Google Scholar 

  10. Manel, L., Ghodrat, E., Essaid, Z., Anjan, B.: Analytical and numerical solutions of the Schrödinger–KdV equation. Pramana 78(1), 59–90 (2012)

    Article  Google Scholar 

  11. Manel, L., Houria, T., Krishnan, E.V., Anjan, B.: Soliton solutions of the long-shortwave equation with power law nonlinearity. J. Appl. Nonlinear Dyn. 1(2), 125–140 (2012)

    Google Scholar 

  12. Ming, S., Zehrong, L., Essaid, Z., Anjan, B.: Singular solitons and bifurcation analysis of quadratic nonlinear Klein–Gordon equation. Appl. Math. Inf. Sci. 7(4), 1333–1340 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Houria, T., Sihon, C., Ahmet, Y., Hayat, T., Aldossary, O.M., Anjan, B.: Bright and dark solitons of the modified complex Ginzburg–Landau with parabolic and dual-power law nonlinearity. Rom. Rep. Phys. 64(2), 367–380 (2012)

    Google Scholar 

  14. Ghodrat, E., Nazila, Y., Houria, T., Ahmet, Y., Anjan, B.: Envelope solitons, periodic waves and other solutions to Boussinesq–Burgers equation. Rom. Rep. Phys. 64(4), 915–932 (2012)

    Google Scholar 

  15. Ablowitz, M.J.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge Univ. Press, Cambridge (1992)

    Google Scholar 

  16. Dickey, L.A.: Soliton Equations and Hamiltonian Systems. World Sci. Press, Singapore (2003)

    Book  MATH  Google Scholar 

  17. Wang, L., Gao, Y.T., Meng, D.X., Gai, X.L., Xu, P.B.: Soliton-shape-preserving and soliton-complex interactions for a (1+1)-dimensional nonlinear dispersive-wave system in shallow water. Nonlinear Dyn. 66, 161–168 (2011)

    Article  MathSciNet  Google Scholar 

  18. Biswas, A., Khalique, C.M.: Stationary solutions for nonlinear dispersive Schrödinger’s equation. Nonlinear Dyn. 63, 623–626 (2011)

    Article  MathSciNet  Google Scholar 

  19. Li, Y., Mu, M.: Baroclinic instability in the generalized Phillips model part I: two-layer model. Adv. Atm. Sci. 13(1), 33–42 (1996)

    Article  MathSciNet  Google Scholar 

  20. Li, Y.: Baroclinic instability in the generalized Phillips model part II: three-layer model. Adv. Atm. Sci. 17(3), 413–432 (2000)

    Article  Google Scholar 

  21. Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  22. Dodd, R.K., Eilkck, J.C., Gibbon, J.D., Moms, H.C.: Solitons and Nonlinear Wave Equations. Academic Press, New York (1982)

    MATH  Google Scholar 

  23. Kamchatnov, A.M., Pavlov, M.V.: Periodic solutions and Whitham equations for the AB system. J. Phys. A, Math. Gen. 28, 3279–3288 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ji, J., Zhang, D.J., Zhang, J.J.: Conservation laws of a discrete soliton. Commun. Theor. Phys. 49, 1105–1108 (2008)

    Article  MathSciNet  Google Scholar 

  26. Abdul, H.K., Houria, T., Anjan, B.: Conservation laws of the Bretherton equation. Appl. Math. Inf. Sci. 7(3), 877–879 (2013)

    Article  MathSciNet  Google Scholar 

  27. Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  28. Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation in Soliton Theory and Its Geometric Applications. Shanghai Sci.-Tech. Pub., Shanghai (2005)

    Google Scholar 

  29. Guo, R., Tian, B., Wang, L.: Soliton solutions for the reduced Maxwell–Bloch system in nonlinear optics via N-fold Darboux transformation. Nonlinear Dyn. 69, 2009–2020 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, Z.Y., Li, L., Li, Z.H., Zhou, G.S.: Modulation instability and solitons on a cw background in an optical fiber with higher-order effects. Phys. Rev. E 67, 026603 (2003)

    Article  MathSciNet  Google Scholar 

  31. Akhmediev, N.N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69(2), 1089–1093 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kamchatnov, A.M.: Nonlinear Periodic Waves and Their Modulations. World Sci. Press, Hong Kong (2000)

    Book  MATH  Google Scholar 

  33. Li, L., Li, Z.H., Li, S.Q., Zhou, G.S.: Modulation instability and solitons on a cw background in inhomogeneous optical fiber media. Opt. Commun. 234, 169–176 (2004)

    Article  Google Scholar 

  34. Wright, O.C.: Homoclinic connections of unstable plane waves of the long-WaveCShort-wave equations. Stud. Appl. Math. 117, 71–93 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Ma, Y.C.: The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60, 43–58 (1979)

    MathSciNet  Google Scholar 

  36. Guo, R., Hao, H.Q.: Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 18, 2426–2435 (2013)

    Article  MathSciNet  Google Scholar 

  37. Guo, R., Hao, H.Q., Zhang, L.L.: Bound solitons and breathers for the generalized coupled nonlinear Schrödinger-Maxwell-Bloch system. Mod. Phys. Lett. B 27(17), 1350130 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to each member of our discussion group for their suggestions. This work has been supported by Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi under Grant No. 2013110, by the National Natural Science Foundation of China under Grant No. 61250011 and by the Natural Science Foundation of Shanxi Province under Grant No. 2012011004-3.

Author information

Authors and Affiliations

  1. School of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China

    Rui Guo, Hui-Qin Hao & Ling-Ling Zhang

Authors
  1. Rui Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui-Qin Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ling-Ling Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui Guo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, R., Hao, HQ. & Zhang, LL. Dynamic behaviors of the breather solutions for the AB system in fluid mechanics. Nonlinear Dyn 74, 701–709 (2013). https://doi.org/10.1007/s11071-013-0998-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-0998-1

Keywords

Search

Navigation