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

In this paper we study the generalized variable-coefficient Gardner equations of the form $u_t+A(t)u^n{u}_x+C(t)u^{2n}{u}_x+B(t)u_{xxx}+Q(t)u=0$. This class broadens out many other equations previously considered: Johnpillai and Khalique (2010), Molati and Ramollo (2012) and Vaneeva, Kuriksha and Sophocleous (2015). Equivalence group of the class under consideration has been constructed which permit an exhaustive study and a simple and clear formulation of the results. Some conservation laws are derived for the nonlinearly self-adjoint equations, based on differential substitutions, and by using the direct method of the multipliers.