Lie Symmetries and Low-Order Conservation Laws of a Family of Zakharov-Kuznetsov Equations in 2 + 1 Dimensions

Abstract

In this work, we study a generalised (2+1) equation of the Zakharov–Kuznetsov (ZK)(m,n,k) equation involving three arbitrary functions. From the point of view of the Lie symmetry theory, we have derived all Lie symmetries of this equation depending on the arbitrary functions. Line soliton solutions have also been obtained. Moreover, we study the low-order conservation laws by applying the multiplier method. This family of equations is rich in Lie symmetries and conservation laws. Finally, when the equation is expressed in potential form, it admits a variational structure in the case when two of the arbitrary functions are linear. In addition, the corresponding Hamiltonian formulation is presented.

Symmetry analysis for the 2 + 1 generalized quantum Zakharov-Kuznetsov equation

“…Hence the qZK is an equation of special interest. The Lie symmetry analysis for the Zakharov-Kuznetsov equation, without the quantum terms, has been studied before in [32]. The Lie symmetries for the fractional differential Zakharov-Kuznetsov(ZK) were found in [33], while for a modified ZK equation the symmetry analysis was performed in [34].…”

Editorial of Papers Published in 2020–2021 in the Mathematics and Symmetry/Asymmetry Section

“…In [1], the authors apply a conditional Lie-Bäcklund symmetry method to investigate the functionally generalized separation of variables for quasi-linear diffusion equations with a source. The paper [2] is devoted to studying a generalised (2 + 1) equation of the Zakharov-Kuznetsov (ZK) (m, n, k) equation involving three arbitrary functions. Lie symmetries and line soliton solutions are derived.…”