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.