Symmetry Solutions and Reductions of a Class of Generalized .2 C 1/-dimensional Zakharov–Kuznetsov Equation

Abstract

In this paper, we study a particular class of the generalized .2 C 1/-Zakharov-Kuznetsov (ZK) equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the underlying equation are derived. We obtain the optimal system of one-dimensional subalgebras of the Lie symmetry algebras of the equation. These subalgebras are then used to reduce the underlying equation to partial differential equations (PDEs) having two independent variables. Furthermore, by studying the reduced PDEs utilizing their symmetry properties, we construct a number of symmetry reductions and exact groupinvariant solutions to the underlying equation. A conserved density is also retrieved by the multiplier approach.

Soliton solutions and chaotic motion of the extended Zakharov-Kuznetsov equations in a magnetized two-ion-temperature dusty plasma

“…18 Some numerical and analytical studies have been conducted on Eq. (3): the explosive and periodic solutions have been obtained, 19 existence and instability of the propagating solitary wave solutions have been simulated numerically, 20 the conserved quantities and one-soliton solutions have been given via the mapping and Ansatz methods and Lie Group analysis, 21,22 the symmetry solutions and reductions have been investigated, 23 and some analytical solutions have been obtained. 24,25 Chaos, which satisfies certain special mathematical criteria and occurs in a deterministic nonlinear system, is a sustained and disorderly looking long-term evolution.…”

Conservation Laws and optimal system of extended quantum Zakharov-Kuznetsov equation

“…The authors [20] employed the reductive perturbation method to formally derive an extended quantum Zakharov-Kuznetsov (extended QZK) equation, which was studied by generalized expansion method [27] and Jacobi elliptic sine and cosine functions [36]. The Lie symmetry approach and the simplest equation method were used to the Zakharov-Kuznetsov modified equal width equation with power law nonlinearity [7] and a class of Generalized (2+1)-dimensional Zakharov-Kuznetsov equation [13].…”

New extended generalized Kudryashov method for solving three nonlinear partial differential equations

“…As mathematical models of the phenomena, the investigations of exact solutions of NLPDEs will help one to understand these phenomena better. In the past several decades, many significant methods for obtaining exact solutions of NLPDEs have been showed, such as the sine-cosine method [3,5,18,44], the modified simple equation method [2,13,27,28,40,43], the soliton ansatz method [6-8, 15, 16, 24, 32], the (G /G)-expansion method [4,12,20,21,42], the generalized Kudryashov method [25,30,41], the modified transformed rational function method [38], the Lie symmetry method [29,34], the travelling wave hypothesis [14,33,39], the extended trial equation method [9-11, 22, 23, 31] and so on.…”