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

Abstract

In this paper we study the generalized variable-coefficient Gardner equations of the form $u_t + A (t) u^n u_x + C(t) u^(2 n) u_x + B (t) u_{xxx} + Q(t) u = 0$ . This class broadens out many other equations previously considered: Johnpillai and Khalique (2010), Molati and Ramollo (2012) and Vaneeva 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.