Conservation Laws and Symmetry Multi-Reductions of Two (2 + 1)-Dimensional Equations: The Zakharov-Kuznetsov (ZK) Equation and a Nonlinear Wave Equation

Abstract

The construction of invariant solutions is a key application of Lie symmetry analysis in studying partial differential equations. The generalised double reduction method, which uses both symmetries and conservation laws of a PDE or system of PDEs, provides a powerful framework for constructing such solutions. This paper contributes to the application of the generalised double
reduction method by analysing two (2 + 1)-dimensional equations: the Zakharov-Kuznetsov (ZK) equation and a nonlinear wave equation. We extend the work of Bokhari et al. [6, 7] on the nonlinear wave equation by performing a second symmetry reduction using previously unused inherited symmetries. For the ZK equation, we identify its Lie point symmetries, construct four conservation laws using the multiplier method, and determine their associated Lie point symmetries. This allows for symmetry reductions using each conservation law. This paper provides a detailed account of the generalised double reduction method, including the exploitation of inherited symmetries at each reduction step.