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