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.
Highlights
In the context of plasma physics, the Zakharov–Kuznetsov equation (ZK) arises to describe ion-sound waves propagating along the magnetic field [1]
We focus our attention on the variational symmetries and conservation laws of the gZK potential
We have considered a generalised ZK equation in (2 + 1)-dimensions depending on three arbitrary functions (6)
Summary
In the context of plasma physics, the Zakharov–Kuznetsov equation (ZK) arises to describe ion-sound waves propagating along the magnetic field [1]. The ZK equation describes the behaviour of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and hot isothermal electrons in a uniform magnetic field. We are interested in studying a generalised ZK equation involving arbitrary functions. We study the following (2 + 1)-dimensional generalised Zakharov–Kuznetsov equation involving three arbitrary functions (gZK). The study of conservation laws of Equation (6) is motivated to determine special cases for the arbitrary functions, f , g and h, with extra conservation laws. Making use of the fact that an equation admits a variational structure if and only if the Frechet derivative of the equation is self-adjoint (i.e., the Helmholtz conditions hold) [15,22], we present the case for the arbitrary functions f , g and h when the potential form of the gZK Equation (6). We determine the variational symmetries of the potential equation admitting the variational structure
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