Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation

•Equivalence groups allow us to reduce the number of arbitrary elements.•Classical symmetries of the equation are obtained for nonlinear terms of any order.•Infinite nontrivial conservation laws of the equation have been constructed.

In this paper we study the generalized variable-coefficient Gardner equations of the form ut+A(t)unux+C(t)u2nux+B(t)uxxx+Q(t)u=0. This class broadens out many other equations previously considered: Johnpillai and Khalique (2010), Molati and Ramollo (2012) and Vaneeva et al. (2015). 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. Some conservation laws are derived for the nonlinearly self-adjoint equations by using a general theorem on conservation laws. We also construct conservation laws by applying the multipliers method.