A group theoretic analysis of the generalised Gardner equation with arbitrary order nonlinear terms

Abstract

Abstract In this paper, we analyse a Gardner equation with dual power law nonlinearity of any order which has been widely investigated due to its applications in quantum field theory, solid state, plasma and fluid physics. A group theoretic approach is used to perform a comprehensive and detailed analysis of the equation. We derive the symmetry generators of the equation in terms of its arbitrary parameters and used them to obtain symmetry reductions and exact solutions. Furthermore, the conservation laws of the equation are derived via the Noether approach after increasing the order and by the use of the multiplier method. The importance of these conservation laws in finding exact solutions is proved via double reduction theory. Most of these solutions are new and contain many known solutions as special cases. These solutions include important soliton solutions and nontrivial solutions in terms of special functions which are meromorphic in the entire complex plane. Since the Gardner equation can model a variety of wave phenomena in plasma, solid state and fluid physics, these solutions possess significant features in the non-linear mechanics aspects of the work. They can also be used as a basis for solving other related model problems and assessing numerical and approximate analytical methods for nonlinear equations describing solitons in wave mechanics.