An Analysis of the Symmetries and Conservation Laws of the Class of Zakharov-Kuznetsov Equations

Abstract

Abstract-In this paper, we study and classify the conservation laws of the ZakharovKuznetsov equations. It is shown that these can be obtained by studying the interplay between symmetry generators and 'multipliers'. This is, particularly, useful for the higher-order multipliers. As a final note, we include Drinfeld-Sokolov-Wilson system to demonstrate the usefulness of the approach to systems of pdes.

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

“…We note that this is significantly different from a more restricted form of the ZK equation discussed in [8] where multipliers of second-order in derivatives and their corresponding conserved flows were obtained. Here, the calculations reveal that we obtain any arbitrary function of y, f .y/, as a multiplier so that f .y/¹u t C a 0 .u n / x C OEb 0 u n .u 2n / xx C c 0 .u n / yy x º D D x T x C D y T y C D t T t in which case the components of the conserved flow are T x D 1 6 .u.uf 00 2f 0 u y / C f .3u 2 C 2.u 2 y C 9u 2 x / C 2u.u yy C 3u xx ///; T y D 1 3 .2f u y u x C u. f 0 u x C 2f u xy //;…”

Soliton solution and conservation laws of the Zakharov equation in plasmas with power law nonlinearity

“…Thus, a knowledge of each multiplier Q leads to a conserved vector determined by, inter alia, a Homotopy operator. See details and references in [2,4].…”

“…There are several nonlinear evolution equations (NLEEs) that appears in various areas of applied mathematics and theoretical physics [1][2][3][4][5][6][7][8][9][10][11][12][13]. These NLEEs are a key to the understanding of various physical phenomena that governs the world today.…”

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

“…By using the multiplier method [20][21][22], we find all local conservation laws admitted by gZK Equation (6). Conservation laws for some special cases of the previous Equations ( 2)-( 4) can be found in [23][24][25]. Consequently, the study of conservation laws of Equation ( 6) is also motivated to determine special cases for the arbitrary functions, f , g and h, with extra conservation laws.…”