Symmetry Reductions and Exact Solutions of a Variable Coefficient (2+1)-Zakharov-Kuznetsov Equation

Abstract

We study the generalized (2+1)-Zakharov-Kuznetsov (ZK) equation of time dependent variable coefficients from the Lie group-theoretic point of view. The Lie point symmetry generators of a special form of the class of equations are derived. We classify the Lie point symmetry generators to obtain the optimal system of onedimensional subalgebras of the Lie symmetry algebras. These subalgebras are then used to construct a number of symmetry reductions and exact group-invariant solutions to the underlying equation.

New Similarity Solutions of a Generalized Variable-Coefficient Gardner Equation with Forcing Term

“…where α 1 (s), α 2 (s), γ(s) and δ(s) are given in Equation (21). Solving the above equations, we get:…”

“…Exact solutions of a nonlinear evolution equation offer direct and valuable insight into the physical aspects of the problem modeled by the equation. Therefore, the search for exact solutions of nonlinear evolution equations has gained much attention in the past decades from the mathematical physics community, and a large number of methods have been proposed, such as the tanh method [1,2], the inverse scattering method [3], the homogeneous balance method [4], the (G /G)-expansion method [5,6], the sine-cosine method [7], the Frobenius integrable decomposition method [8], the improved Exp-function method [9], the generalized Kudryashov method [10], the local fractional Riccati differential equation method [11], the Hirota bilinear method [12][13][14], the Darboux transformation method [15,16] and the group methods [17][18][19][20][21][22][23][24][25]. As stated in [26], symmetry is the key to solving differential equations.…”

Symmetry reductions of the ( 3 + 1 ) $(3+1)$ -dimensional modified Zakharov–Kuznetsov equation

“…The Lie-group method, originally proposed by Sophus Lie, is a classical method to determine the symmetry reduction of partial differential equations (PDEs) [13][14][15][16]. During the past several decades, there have been many extensions of the Lie-group method such as the nonclassical Lie group method [17], the CK direct method [18], the direct symmetry method [19], and so on [20][21][22][23][24]. Among them, the direct symmetry method is an effective approach for seeking symmetry reductions.…”

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

“…The Lie method determines all the Lie symmetries that a given PDE admits [12][13][14][15]. Moreover, for different cases of the ZK Equation ( 2)-( 4), the Lie method has been shown as a useful tool to get exact solutions, including soliton solutions, cnoidal waves and travelling wave solutions [16][17][18][19]. In Section 3, line soliton solutions have been obtained.…”