Abstract
In this paper, a simple way to construct exact solutions by using equivalence transformations is shown. We consider a generalized variable-coefficient Gardner equation from the point of view of Lie symmetries in partial differential equations. We obtain the continuous equivalence transformations of the equation in order to reduce the number of arbitrary functions and give a clearer formulation of the results. Furthermore, we calculate Lie symmetries of the reduced equation. Then, we determine the similarity variables and the similarity solutions which allow us to reduce our equation into an ordinary differential equation. Finally, we obtain some exact travelling wave solutions of the equation by using the simplest equation method.
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Acknowledgements
We warmly thank the referee his valuable comments and suggestions. The authors acknowledge the financial support from Junta de Andalucía group FQM–201, Universidad de Cádiz. The first author expresses his sincere gratitude to the Plan Propio de Investigación 2013 de la Universidad de Cádiz and the Comisión Académica del Programa de Doctorado en Matemáticas de la Universidad de Cádiz for their support. The second author also acknowledges the support of DGICYT project MTM2009-11875 with the participation of FEDER.
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de la Rosa, R., Bruzón, M.S. (2016). Travelling Wave Solutions of a Generalized Variable-Coefficient Gardner Equation. In: Ortegón Gallego, F., Redondo Neble, M., Rodríguez Galván, J. (eds) Trends in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-32013-7_23
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DOI: https://doi.org/10.1007/978-3-319-32013-7_23
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