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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:479

Article

Okeke, J. E.¹  Narain, R.¹  Govinder, K. S.¹ 

[1] Univ KwaZulu Natal, Sch Math Stat & Comp Sci, Private Bag X54001, ZA-4000 Durban, South Africa

关键词: Double reduction theory;    Conservation laws;    Lie symmetry method;    Riccati equation method;   

DOI  :  10.1016/j.jmaa.2019.07.036

来源: Elsevier

【 摘 要 】

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 nonlinear 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free