Abstract
In this paper, a generalized Gardner equation with nonlinear terms of any order has been analyzed from the point of view of group transformations and conservation laws. The generalized Gardner equation appears in many areas of physics and it is widely used to model a great variety of wave phenomena in plasma and solid state. By using the direct method of the multipliers, we have obtained an exhaustive classification of all low-order conservation laws which the generalized Gardner equation admits. Then, taking into account these conserved vectors we have determined the associated potential systems and we have searched for potential symmetries of the equation. Furthermore, we have determined and examined its first-level and second-level potential systems. From the first-level potential system we have found two new nonlocal conserved vectors.
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References
Anco, S.C.: On the incompleteness of Ibragimov’s conservation law theorem and its equivalence to a standard formula using symmetries and adjoint-symmetries. Symmetry 9, 33 (28 pp.) (2017)
Anco, S.C.: Generalization of Noether’s theorem in modern form to non-variational partial differential equations. In: Fields Institute Communications: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, vol. 79. Springer, New York (2017)
Anco, S.C., Bluman, G.W.: Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78, 2869–2873 (1997)
Anco, S.C., Bluman, G.W.: Direct construction method for conservation laws of partial differential equations Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13, 545–566 (2002)
Anco, S.C., Bluman, G.W.: Direct construction method for conservation laws of partial differential equations Part II: general treatment. Eur. J. Appl. Math. 13, 567–585 (2002)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)
Bluman, G.W., Reid, G.J., Kumei, S.: New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811 (1988)
Bluman, G.W., Cheviakov, A., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2009)
Brauer, K.: The Korteweg-de Vries equation: history, exact solutions, and graphical representation. University of Osnabrück, Osnabrück (2000). http://math.arizona.edu/~gabitov/teaching/141/math_485/KDV.pdf
de la Rosa, R., Bruzón, M.S.: On the classical and nonclassical symmetries of a generalized Gardner equation. Appl. Math. Nonlinear Sci. 1, 263–272 (2016)
de la Rosa, R., Bruzón, M.S.: Travelling wave solutions of a generalized variable-coefficient Gardner equation. In: Trends in Differential Equations and Applications. SEMA SIMAI Springer Series, pp. 405–417. Springer, Cham (2016)
de la Rosa, R., Bruzón, M.S.: On conservation laws of a generalized Gardner equation. In: Proceedings of XXV CEDYA/XV CMA, pp. 244–248 (2017)
de la Rosa, R., Gandarias, M.L., Bruzón, M.S.: Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation. Commun. Nonlinear Sci. Numer. Simul. 40, 71–79 (2016)
Dimas, S., Freire, I.L.: Study of a fifth order PDE using symmetries. Appl. Math. Lett. 69, 121–125 (2017)
Gandarias, M.L., Khalique, C.M.: Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations. Commun. Nonlinear Sci. Numer. Simul. 32, 114–121 (2016)
Gandarias, M.L., de la Rosa, R., Rosa, M.: Conservation laws for a strongly damped wave equation. Open Phys. 15, 300–305 (2017)
Garrido, T.M., Kasatkin, A.A., Bruzón, M.S., Gazizov, R.K.: Lie symmetries and equivalence transformations for the Barenblatt-Gilman model. J. Comput. Appl. Math. 318, 253–258 (2017)
Molati, M., Ramollo, M.P.: Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids. Commun. Nonlinear Sci. Numer. Simul. 17, 1542–1548 (2012)
Olver, P.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Ovsyannikov, L.V.: Group Analysis of Differential Equations. Academic, New York (1982)
Recio, E., Anco, S.C.: Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations. J. Math. Anal. Appl. 452, 1229–1261 (2017)
Rosa, M., Gandarias, M.L.: Multiplier method and exact solutions for a density dependent reaction-diffusion equation. Appl. Math. Nonlinear Sci. 1(2), 311–320 (2016)
Torrisi, M., Tracinà, R.: Exact solutions of a reaction-diffusion system for Proteus mirabilis bacterial colonies. Nonlinear Anal. Real World Appl. 12, 1865–1874 (2011)
Tracinà, R., Gandarias, M.L., Bruzón, M.S., Torrisi, M.: Nonlinear self-adjointness, conservation laws, exact solutions of a system of dispersive evolution equations. Commun. Nonlinear Sci. Numer. Simul. 19, 3036–3043 (2014)
Vaneeva, O., Kuriksha, O., Sophocleous, C.: Enhanced group classification of Gardner equations with time-dependent coefficients. Commun. Nonlinear Sci. Numer. Simul. 22, 1243–1251 (2015)
Acknowledgements
The authors gratefully acknowledge the financial support from the Universidad Politécnica de Cartagena. The authors also acknowledge the financial support from Junta de Andalucía group FQM-201 and they express their sincere gratitude to the Plan Propio de Investigación de la Universidad de Cádiz.
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la Rosa, R.d., Garrido, T.M., Bruzón, M.S. (2019). Conservation Laws and Potential Symmetries for a Generalized Gardner Equation. In: García Guirao, J., Murillo Hernández, J., Periago Esparza, F. (eds) Recent Advances in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-00341-8_7
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