Exact Explicit Solutions and Conservation Laws for a Coupled Zakharov-Kuznetsov System

We study a coupled Zakharov-Kuznetsov system, which is an 
extension of a coupled Korteweg-de Vries system in the sense of the Zakharov-Kuznetsov equation. Firstly, we obtain some exact solutions of the coupled 
Zakharov-Kuznetsov system using the simplest equation method. Secondly, 
the conservation laws for the coupled Zakharov-Kuznetsov system will be 
constructed by using the multiplier approach.


Introduction
It is well known that the two-dimensional generalizations of the Korteweg-de Vries (KdV) equation are the Kadomtsev-Petviashivili (KP) equation and the Zakharov-Kuznetsov (ZK) equation.The ZK equation governs the behaviour of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [1].
In [2] a new hierarchy of nonlinear evolution equations was derived, and one particular system of equations where , , , and  are constants, was later studied by [3].This coupled KdV system (3a), (3b), and (3c) was extended to the new coupled ZK system in the sense of the ZK Equation ( 2) in [1], and travelling wave solutions were determined using the extended tanhcoth method and sech method.
In the last few decades, several powerful methods have been introduced in the literature, which can be used to find exact solutions of nonlinear differential equations arising from physical problems.These methods include the inverse scattering transform method [4], the Darboux transformation [5], the Hirota's bilinear method [6], the Jacobi elliptic function expansion method [7,8], the multiple-exp method [9], the sine-cosine method [10], the Lie symmetry method [11,12], and the (  /)-expansion method [13].

Exact Solutions Using Simplest Equation Method
In this section we employ the simplest equation method [14,15] and obtain some exact explicit solutions of (4a), (4b), and (4c).The simplest equations that will be used in this paper are the Bernoulli and Riccati equations.It is well known that their solutions can be written in elementary functions.See, for example, [19].By using the transformation where   ,  = 1, . . ., 4, are constants, the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c) transforms to a thirdorder coupled system of nonlinear ordinary differential equations (ODEs) We now present the simplest equation method for a system of three ODEs.Consider the solutions of (6a), (6b), and (6c) in the form where () satisfies the Bernoulli or Riccati equation,  is a positive integer that can be determined by balancing procedure [15], and   ,   , and   ( = 0, 1, . . ., ) are parameters to be determined.
The Bernoulli equation we consider in this paper is where  and  are constants.Its solution can be written as For the Riccati equation where , , and  are constants, we will use the solutions where  2 =  2 − 4.