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An Overview of the Generalized Gardner Equation: Symmetry Groups and Conservation Laws

  • M. S. Bruzón
  • M. L. Gandarias
  • R. de la Rosa
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 22)

Abstract

In this paper we study the generalized variable-coefficient Gardner equations of the form u t  + A(t)f(u)u x  + C(t)f(u)2u x  + B(t)u xxx  + Q(t)F(u) = 0. This family of equations includes many equations considered in the literature. Some conservation laws are derived by applying the multipliers method. The use of the equivalence group of this class allows us to perform an exhaustive study and a simple and clear formulation of the results. We study the equation from the point of view of Lie symmetries in partial differential equations. Finally, we calculate exact travelling wave solutions of the equation by using the simplest equation method.

Notes

Acknowledgements

The authors acknowledge the financial support from Junta de Andalucía group FQM-201, they express their sincere gratitude to the Plan Propio de Investigación and project PR2016-097 de la Universidad de Cádiz. Bruzón and Gandarias are also grateful to the Organizing Committee of NSC-2016 for giving them the chance to participate in the conference.

References

  1. 1.
    Abdel-Gawad, H. I., & Tantawy, M. (2014). Exact solutions of the Shamel-Korteweg-de Vries equation with time dependent coefficients. Information Sciences Letters, 3(3), 103–109.CrossRefGoogle Scholar
  2. 2.
    Adem, K. R., & Khalique, C. M. (2012). Exact solutions and conservation laws of Zakharov-Kuznetsov modified equal width equation with power law nonlinearity. Nonlinear Analysis: Real World Applications, 13, 1692–1702.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anco, S. C. (2017). Generalization of Noether’s theorem in modern form to non-variational partial differential equations. In Recent progress and modern challenges in applied mathematics, modeling and computational science. Fields institute communications (pp. 79–130). New York: Springer.CrossRefGoogle Scholar
  4. 4.
    Anco, S. C. (2017). On the incompleteness of Ibragimov’s conservation law theorem and its equivalence to a standard formula using symmetries and adjoint symmetries. Symmetry, 9(33), 1–28.MathSciNetGoogle Scholar
  5. 5.
    Anco, S. C., Avdonina, E. D., Gainetdinova, A., Galiakberova, L. R., Ibragimov, N. H., & Wolf, T. (2016). Symmetries and conservation laws of the generalized Krichever-Novikov equation. Journal of Physics A: Mathematical and Theoretical, 49, 105201–105230.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Anco, S. C., & Bluman, G. (2002). Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications. European Journal of Applied Mathematics, 13, 545–566.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Anco, S. C., & Bluman, G. (2002). Direct construction method for conservation laws of partial differential equations. Part II: General treatment. European Journal of Applied Mathematics, 13, 567–585.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Avdonina, E. D., & Ibragimov, N. H. (2013). Conservation laws and exact solutions for nonlinear diffusion in anisotropic media. Communications in Nonlinear Science and Numerical Simulation, 18, 2595–2603.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bozhkov, Y., Dimas, S., & Ibragimov, N. H. (2013). Conservation laws for a coupled variable-coefficient modified Korteweg-de Vries system in a two-layer fluid model. Communications in Nonlinear Science and Numerical Simulation, 18, 1127–1135.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bruzón, M. S., & de la Rosa, R. (2014). Analysis of the symmetries and conservation laws of a Gardner equation. In AIP Conference Proceedings of ICNAAM, Rhodes, Greece.Google Scholar
  11. 11.
    Bruzón, M. S., Gandarias, M. L., & de la Rosa, R. (2014). Conservation laws of a family reaction-diffusion-convection equations. In Localized excitations in nonlinear complex systems. Nonlinear systems and complexity (Vol. 7). Basel: Springer International Publishing.Google Scholar
  12. 12.
    Bruzón, M. S., Garrido, T. M., & de la Rosa, R. (2016). Conservation laws and exact solutions of a generalized Benjamin-Bona-Mahony-Burgers equation. Chaos, Solitons and Fractals, 89, 578–583.MathSciNetCrossRefGoogle Scholar
  13. 13.
    de la Rosa, M. L., & Bruzón, M. S. (2016). On the classical and nonclassical symmetries of a generalized Gardner equation. Applied Mathematics and Nonlinear Sciences, 1(1), 263–272.CrossRefGoogle Scholar
  14. 14.
    de la Rosa, R., Gandarias, M. L., & Bruzón, M. S. (2016). On symmetries and conservation laws of a Gardner equation involving arbitrary functions. Applied Mathematics and Computation, 290, 125–134.MathSciNetCrossRefGoogle Scholar
  15. 15.
    de la Rosa, R., Gandarias, M. L., & Bruzón, M. S. (2016). Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation. Preprint. Communications in Nonlinear Science and Numerical Simulation, 40, 71–79.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Freire, I. L., & Sampaio, J. C. S. (2014). On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models. Communications in Nonlinear Science and Numerical Simulation, 19, 350–360.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gandarias, M. L. (2011). Weak self-adjoint differential equations. Journal of Physics A: Mathematical and Theoretical, 44, 262001 (6 pp.).CrossRefGoogle Scholar
  18. 18.
    Hong, B., & Lu, D. (2012). New exact solutions for the generalized variable-coefficient Gardner equation with forcing term. Applied Mathematics and Computation, 219, 2732–2738.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hubert, M. B., Betchewe, G., Doka, S. Y., & Crepin, K. T. (2014). Soliton wave solutions for the nonlinear transmission line using the Kudryashov method and the \(\left (G'/G\right )\)-expansion method. Applied Mathematics and Computation, 239, 299–309.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ibragimov, N. H. (2006). The answer to the question put to me by LV Ovsiannikov 33 years ago. Archives of ALGA, 3, 53–80.Google Scholar
  21. 21.
    Ibragimov, N. H. (2007). A new conservation theorem. Journal of Mathematical Analysis and Applications, 333, 311–328.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ibragimov, N. H. (2007). Quasi-self-adjoint differential equations. Archives of ALGA, 4, 55–60.Google Scholar
  23. 23.
    Ibragimov, N. H. (2011). Nonlinear self-adjointness and conservation laws. Journal of Physics A: Mathematical and Theoretical, 44, 432002 (8 pp.).CrossRefGoogle Scholar
  24. 24.
    Ibragimov, N. K. (1985). Transformation groups applied to mathematical physics. Dordrecht: Reidel.CrossRefGoogle Scholar
  25. 25.
    Johnpillai, A. G., & Khalique, C. M. (2010). Group analysis of KdV equation with time dependent coefficients. Applied Mathematics and Computation, 216, 3761–3771.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Johnpillai, A. G., & Khalique, C. M. (2011). Conservation laws of KdV equation with time dependent coefficients. Communications in Nonlinear Science and Numerical Simulation, 16, 3081–3089.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kudryashov, N. A. (2005). Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons and Fractals, 24, 1217–1231.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kudryashov, N. A. (2010). Meromorphic solutions of nonlinear ordinary differential equations. Communications in Nonlinear Science and Numerical Simulation, 15, 2778–2790.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kudryashov, N. A. (2015). Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a source. Applied Mathematics Letters, 41, 41–45.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kudryashov, N. A., & Loguinova, N. B. (2008). Extended simplest equation method for nonlinear differential equations. Applied Mathematics and Computation, 205, 396–402.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Molati, M., & Ramollo, M. P. (2012). Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids. Communications in Nonlinear Science and Numerical Simulation, 17, 1542–1548.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Olver, P. (1993). Applications of Lie groups to differential equations. New York: Springer.CrossRefGoogle Scholar
  33. 33.
    Ovsyannikov, L. V. (1982). Group analysis of differential equations. New York: Academic.zbMATHGoogle Scholar
  34. 34.
    Tracinà, R. (2014). On the nonlinear self-adjointness of the Zakharov-Kuznetsov equation. Communications in Nonlinear Science and Numerical Simulation, 19, 337–382.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tracinà, R. (2015). Nonlinear self-adjointness: a criterion for linearization of PDEs. Journal of Physics A: Mathematical and Theoretical, 48, 06FT01 (10 pp.).MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tracinà, R., Bruzón, M. S., Gandarias, M. L., & Torrisi, M. (2014). Nonlinear self-adjointness, conservation laws, exact solutions of a system of dispersive evolution equations. Communications in Nonlinear Science and Numerical Simulation, 19, 3036–3043.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Tracinà, R., Freire, I. L., & Torrisi, M. (2016). Nonlinear self-adjointness of a class of third order nonlinear dispersive equations. Communications in Nonlinear Science and Numerical Simulation, 32, 225–233.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wang, G. W., Liu, X. G., & Zhang, Y. (2013). Symmetry reduction, exact solutions and conservation laws of a new fifth-order nonlinear integrable equation. Communications in Nonlinear Science and Numerical Simulation, 18, 2313–2320.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Wei, L. (2015). Conservation laws for a modified lubrication equation. Nonlinear Analysis: Real World Applications, 26, 44–55.MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wolf, T. (1993). An efficiency improved program LIEPDE for determining Lie-symmetries of PDEs. In Proceedings of Modern Group Analysis: Advances Analytical and Computational Methods in Mathematical Physics (pp. 377–385).CrossRefGoogle Scholar
  41. 41.
    Zhang, L. H., Dong, L. H., & Yan, L. M. (2008). Construction of non-travelling wave solutions for the generalized variable-coefficient Gardner equation. Applied Mathematics and Computation, 203, 784–791.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • M. S. Bruzón
    • 1
  • M. L. Gandarias
    • 1
  • R. de la Rosa
    • 1
  1. 1.Departamento de MatemáticasUniversidad de CádizCádizSpain

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