Nonlinear Dynamics

, Volume 84, Issue 1, pp 135–141 | Cite as

Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping

  • R. de la RosaEmail author
  • M. L. Gandarias
  • M. S. Bruzón
Original Paper


A fifth-order KdV equation with time-dependent coefficients and linear damping has been studied. Symmetry groups have several different applications in the context of nonlinear differential equations; for instance, they can be used to determine conservation laws. We obtain the symmetries of the model applying Lie’s classical method. The choice of some arbitrary functions of the equation by the equivalence transformation enhances the study of Lie symmetries of the equation. We have determined the subclasses of the equation which are nonlinearly self-adjoint. This allow us to obtain conservation laws by using a theorem proved by Ibragimov which is based on the concept of adjoint equation for nonlinear differential equations.


Classical symmetries Equivalence transformations Partial differential equations Conservation laws 

Mathematics Subject Classification

35C07 35Q53 



We would like to thank the Editor and Referees for their timely and valuable comments and suggestions. The authors acknowledge the financial support from Junta de Andalucia group FQM-201. The first author express his sincerest gratitude to the Universidad Politécnica de Cartagena for supporting him. The second and third authors also acknowledge the support of DGICYT Project MTM2009-11875 with the participation of FEDER.


  1. 1.
    Biswas, A., et al.: Solitons and conservation laws of KleinGordon equation with power law and log law nonlinearities. Nonlinear Dyn. 73, 2191–2196 (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bruzón, M.S., Gandarias, M.L., Ibragimov, N.H.: Self-adjoint sub-classes of generalized thin film equations. J. Math. Anal. Appl. 357, 307–313 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruzón, M.S., Gandarias, M.L., de la Rosa, R.: Conservation Laws of a Family of Reaction-Diffusion-Convection Equations in Localized Excitations in Nonlinear Complex Systems. Springer, New York (2014)zbMATHGoogle Scholar
  4. 4.
    de la Rosa, R., Gandarias, M.L., Bruzón, M.S.: On symmetries and conservation laws of a Gardner equation involving arbitrary functions. Appl. Math. Comput. (in press)Google Scholar
  5. 5.
    Freire, I.L., Sampaio, J.C.S.: Nonlinear self-adjointness of a generalized fifth-order KdV equation. J. Phys. A Math. Theory 45, 032001–032007 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gandarias, M.L., Bruzón, M.S.: Some conservation laws for a forced KdV equation. Nonlinear Anal. Real World Appl. 13, 2692–2700 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gupta, R.K., Bansal, A.: Similarity reductions and exact solutions of generalized Bretherton equation with time-dependent coefficients. Nonlinear Dyn. 71, 1–12 (2013)Google Scholar
  8. 8.
    Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ibragimov, N.H.: Nonlinear self-adjointness and conservation laws. J. Phys. A Math. Theory 44, 432002–432010 (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Krishnan, E.V., et al.: A study of shallow water waves with Gardner’s equation. Nonlinear Dyn. 66, 497–507 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, H., Yan, F., Xu, C.: The bifurcation and exact travelling wave solutions of (1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity. Nonlinear Dyn. 67, 465–473 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Olver, P.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ovsyannikov, L.V.: Group Analysis of Differential Equations. Academic, New York (1982)zbMATHGoogle Scholar
  14. 14.
    Razborova, P., Kara, A.H., Biswas, A.: Additional conservation laws for Rosenau-KdV-RLW equation with power law nonlinearity by Lie symmetry. Nonlinear Dyn. 79, 743–748 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Senthilvelan, M., Torrisi, M., Valenti, A.: Equivalence transformations and differential invariants of a generalized nonlinear Schrödinger equation. J. Phys. A Math. Gen. 39, 3703–3713 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Vaneeva, O., et al.: Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities. J. Math. Anal. Appl. 330, 1363–1386 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, G.W., Xu, T.Z.: Invariant analysis and exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation by Lie group analysis. Nonlinear Dyn. 76, 571–580 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zedan, H.A., Aladrous, E., Shapll, S.: Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations. Nonlinear Dyn. 74, 1145–1151 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • R. de la Rosa
    • 1
    Email author
  • M. L. Gandarias
    • 1
  • M. S. Bruzón
    • 1
  1. 1.Department of MathematicsUniversity of CádizCádizSpain

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