Abstract
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.
1. 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 equationswhere , , , and are constants, was later studied by [3]. This coupled KdV system (3a), (3b), and (3c) was extended to the new coupled ZK systemin the sense of the ZK Equation (2) in [1], and travelling wave solutions were determined using the extended tanh-coth 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].
The purpose of this paper is to employ the simplest equation method [14, 15] to obtain some exact explicit solutions of the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c). Furthermore, we derive conservation laws for (4a), (4b), and (4c) using the multiplier approach [16–18].
2. 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 , , are constants, the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c) transforms to a third-order 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 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 .
2.1. Solutions of (4a), (4b), and (4c) Using the Bernoulli Equation as the Simplest Equation
The balancing procedure yields . Thus, the solutions of (6a), (6b), and (6c) are of the formSubstituting (12a), (12b), and (12c) into (6a), (6b), and (6c) and making use of the Bernoulli equation (8) and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , , and . Solving this system of algebraic equations, with the aid of Mathematica, one possible set of values of , , and is
As a result, a solution of (4a), (4b), and (4c) iswhere .
2.2. Solutions of (4a), (4b), and (4c) Using Riccati Equation as the Simplest Equation
The balancing procedure yields and so the solutions of (6a), (6b), and (6c) are of the formSubstituting (15a), (15b), and (15c) into (6a), (6b), and (6c) and using (10), we obtain an algebraic system of equations in terms of , , and by equating all coefficients of the functions to zero. Solving the resultant system, one possible set of values is
Consequently, the solutions of (4a), (4b), and (4c) arewhere .
3. Conservation Laws of (4a), (4b), and (4c)
In this section we derive conservation laws for the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c). The multiplier approach will be used. For details the reader is referred to [11, 16–18].
In our case we obtain multipliers [17] of the form and corresponding to the above multipliers we then obtain the following conserved vectors [17] of (4a), (4b), and (4c): It should be noted that due to the presence of the arbitrary function in the multipliers there are infinitely many conservation laws for the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c).
4. Concluding Remarks
In this paper we obtained some exact solutions of the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c) by the aid of the simplest equation method. The solutions obtained are solitary waves. Moreover, the conservation laws for the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c) were also derived by using the multiplier approach.