Skip to main content
SpringerLink
Log in

Solitary and extended waves in the generalized sinh-Gordon equation with a variable coefficient

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

Abstract

New solitary and extended wave solutions of the generalized sinh-Gordon (SHG) equation with a variable coefficient are found by utilizing the self-similar transformation between this equation and the standard SHG equation. Two arbitrary self-similar functions are included in the known solutions of the standard SHG equation, to obtain exact solutions of the generalized SHG equation with a specific variable coefficient. Our results demonstrate that the solitary and extended waves of the variable-coefficient SHG equation can be manipulated and controlled by a proper selection of the two arbitrary self-similar functions.

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

Similar content being viewed by others

References

  1. Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: from Fibers to Photonic Crystals. Academic Press, New York (2003)

    Google Scholar 

  2. Malomed, B.A., Mihalache, D., Wise, F., Torner, L.: Spatiotemporal optical solitons. J. Opt. B, Quantum Semiclass. Opt. 7, R53–R72 (2005)

    Article  Google Scholar 

  3. Chern, S.S.: Geometrical interpretation of the sinh-Gordon equation. Ann. Pol. Math. 39, 74–80 (1980)

    MathSciNet  Google Scholar 

  4. Wazwaz, A.: Travelling wave solutions for combined and double combined sine–cosine-Gordon equations by the variable separated ODE method. Appl. Math. Comput. 177, 755–766 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chelnokov, V.E., Zeitlin, M.G.: The elliptic solution of the sinh-Gordon equation. Phys. Lett. A 99, 147–149 (1983)

    Article  MathSciNet  Google Scholar 

  6. Liu, C.S.: Representations and classification of traveling wave solutions to sinh-Gordon equation. Commun. Theor. Phys. 49, 153–158 (2008)

    Article  Google Scholar 

  7. Fu, Z.T., Liu, S.K., Liu, S.D.: Exact Jacobian elliptic function solutions to sinh-Gordon equation. Commun. Theor. Phys. 45, 55–60 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhang, H.: New exact solutions for the sinh-Gordon equation. Chaos Solitons Fractals 28, 489–496 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Wazwaz, A.: Exact solutions of the generalized sine-Gordon and the generalized sinh-Gordon equations. Chaos Solitons Fractals 28, 127–135 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kundu, A.: Shape changing and accelerating solitons in integrable variable mass sine-Gordon model. Phys. Rev. Lett. 99, 154101 (2007)

    Article  Google Scholar 

  11. Belić, M., Petrović, N., Zhong, W.P., Xie, R.H., Chen, G.: Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation. Phys. Rev. Lett. 101, 123904 (2008)

    Article  Google Scholar 

  12. Wu, L., Li, L., Zhang, J.F., Mihalache, D., Malomed, B.A., Liu, W.M.: Exact solutions of the Gross–Pitaevskii equation for stable vortex modes in two-dimensional Bose–Einstein condensates. Phys. Rev. A 81, 061805 (2010)

    Article  Google Scholar 

  13. Tian, Q., Wu, L., Zhang, J.F., Malomed, B.A., Mihalache, D., Liu, W.M.: Exact soliton solutions and their stability control in the nonlinear Schrödinger equation with spatiotemporally modulated nonlinearity. Phys. Rev. E 83, 016602 (2011)

    Article  MathSciNet  Google Scholar 

  14. Zhong, W.P., Belić, M.R., Mihalache, D., Malomed, B.A., Huang, T.W.: Varieties of exact solutions for the (2+1)-dimensional nonlinear Schrödinger equation with the trapping potential. Rom. Rep. Phys. 64, 1399–1412 (2012)

    Google Scholar 

  15. Malomed, B.A.: Soliton Management in Periodic Systems. Springer, New York (2006)

    MATH  Google Scholar 

  16. Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)

    Article  Google Scholar 

  17. Zhong, W.P., Xie, R.H., Belić, M., Petrović, N., Chen, G.: Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients. Phys. Rev. A 78, 023821 (2008)

    Article  Google Scholar 

  18. Zhong, W.P., Belić, M.: Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with harmonic potential and variable coefficients. Phys. Rev. E 82, 047601 (2010)

    Article  Google Scholar 

  19. Zhong, W.P., Belić, M., Huang, T.: Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients. Phys. Rev. E 87, 065201 (2013)

    Article  Google Scholar 

  20. Zhong, W.P., Belić, M., Huang, T.: Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity. J. Opt. Soc. Am. B, Opt. Phys. 30, 1276–1283 (2013)

    Article  Google Scholar 

  21. Zhong, W.P., Belić, M., Huang, T.: Two-Dimensional accessible solitons in PT-symmetric potentials. Nonlinear Dyn. 70, 2027–2034 (2012)

    Article  Google Scholar 

  22. Zhong, W.P., Belić, M., Assanto, G., Malomed, B.A., Huang, T.: Self-trapping of scalar and vector dipole solitary waves in Kerr media. Phys. Rev. A 83, 043833 (2011)

    Article  Google Scholar 

  23. Zhong, W.P., Belić, M.: Resonance soliton clusters by azimuthal modulation in self-focusing and self-defocusing materials. Nonlinear Dyn. 73, 2091–2102 (2013)

    Article  Google Scholar 

  24. Zhong, W.P., Belić, M., Xia, Y.: Special soliton structures in the (2+1)-dimensional nonlinear Schrödinger equation with radially variable diffraction and nonlinearity coefficients. Phys. Rev. E 83, 036603 (2011)

    Article  Google Scholar 

  25. Ma, W.X., Lee, J.H.: A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation. Chaos Solitons Fractals 42, 1356 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ma, W.X., Zhu, Z.: Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ma, W.X., Phys, J.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys., Conf. Ser. 411, 012021 (2013)

    Article  Google Scholar 

  28. Eichelkraut, T.J., Siviloglou, G.A., Besieris, I.M., Christodoulides, D.N.: Oblique Airy wave packets in bidispersive optical media. Opt. Lett. 35, 3655 (2010)

    Article  Google Scholar 

  29. Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, Boston (1997)

    Google Scholar 

  30. Chow, K.W., Grimshaw, R.H.J., Ding, E.: Interactions of breathers and solitons in the extended Korteweg–de Vries equation. Wave Motion 43, 158–166 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ferreira, L.A., Piette, B., Zakrzewski, W.J.: Wobbles and other kink-breather solutions of the sine Gordon model. Phys. Rev. E 77, 036613 (2008)

    Article  Google Scholar 

  32. Yang, Z., Zhong, W.P., Belić, M.: Breather solutions to the nonlinear Schrödinger equation with variable coefficients and a linear potential. Phys. Scr. 86, 015402 (2012)

    Article  Google Scholar 

  33. Chen, Z.G., McCarthy, K.: Spatial soliton pixels from partially coherent light. Opt. Lett. 27, 2019–2021 (2002)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under grant No. 61275001, Ministry of Education and Science of the Republic of Serbia, under projects OI 171033. The work at the Texas A&M University at Qatar was supported by the NPRP 09-462-1-074 project with the Qatar National Research Fund (a member of the Qatar Foundation).

Author information

Authors and Affiliations

  1. Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province, 528300, Shunde, China

    Wei-Ping Zhong

  2. Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar

    Wei-Ping Zhong & Milivoj R. Belić

  3. Institute of Physics, University of Belgrade, P.O.B. 68, 11001, Belgrade, Serbia

    Milivoj R. Belić & Milan S. Petrović

Authors
  1. Wei-Ping Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milivoj R. Belić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milan S. Petrović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-Ping Zhong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhong, WP., Belić, M.R. & Petrović, M.S. Solitary and extended waves in the generalized sinh-Gordon equation with a variable coefficient. Nonlinear Dyn 76, 717–723 (2014). https://doi.org/10.1007/s11071-013-1162-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-1162-7

Keywords

Search

Navigation