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

Authors: Chaudry Masood Khalique
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 1. Introduction

It is well known that the two-dimensional generalizations of the Korteweg-de Vries (KdV) equation(1)ut+auux+uxxx=0
are the Kadomtsev-Petviashivili (KP) equation and the Zakharov-Kuznetsov (ZK) equation. The ZK equation
(2)ut+auux+b(uxx+uyy)x=0
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(3a)ut=βuxxx+α(uv)x+γ(vw)x,(3b)vt=βvxxx+λ(wu)x,(3c)wt=βwxxx+λ(uv)x,where α, β, γ, and λ are constants, was later studied by [3]. This coupled KdV system (3a), (3b), and (3c) was extended to the new coupled ZK system(4a)ut-α(uv)x-γ(vw)x-β(uxx+uyy)x=0,(4b)vt-λ(wu)x-β(vxx+vyy)x=0,(4c)wt-λ(uv)x-β(wxx+wyy)x=0,in 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 (G′/G)-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(5)z=k1t+k2x+k3y+k4,
where ki, i=1,…,4, are constants, the coupled Zakharov-Kuznetsov system (4a), (4b), and (4c) transforms to a third-order coupled system of nonlinear ordinary differential equations (ODEs)(6a)βk23E′′′(z)+βk32k2E′′′(z)+αk2F(z)E′(z)-k1E′(z)+αk2E(z)F′(z)+γk2G(z)F′(z)+γk2F(z)G′(z)=0,(6b)βk23F′′′(z)+βk32k2F′′′(z)+λk2G(z)E′(z)+λk2E(z)G′(z)-k1F′(z)=0,(6c)βk23G′′′(z)+βk32k2G′′′(z)+λk2F(z)E′(z)+λk2E(z)F′(z)-k1G′(z)=0.We now present the simplest equation method for a system of three ODEs. Consider the solutions of (6a), (6b), and (6c) in the form
(7)E(z)=∑i=0M‍Ai(H(z))i,F(z)=∑i=0M‍Bi(H(z))i,G(z)=∑i=0M‍Ci(H(z))i,
where H(z) satisfies the Bernoulli or Riccati equation, M is a positive integer that can be determined by balancing procedure [15], and Ai, Bi, and Ci(i=0,1,…,M) are parameters to be determined.The Bernoulli equation we consider in this paper is(8)H′(z)=aH(z)+bH2(z),
where a and b are constants. Its solution can be written as
(9)H(z)=a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}.
For the Riccati equation
(10)H′(z)=aH2(z)+bH(z)+c,
where a, b, and c are constants, we will use the solutions
(11)H(z)=-b2a-θ2atanh[12θ(z+C)],H(z)=-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2),
where θ2=b2-4ac.


### 2.1. Solutions of (4a), (4b), and (4c) Using the Bernoulli Equation as the Simplest Equation

The balancing procedure yieldsM=2. Thus, the  solutions of (6a), (6b), and (6c) are of the form(12a)E(z)=A0+A1H+A2H2,(12b)F(z)=B0+B1H+B2H2,(12c)G(z)=C0+C1H+C2H2.Substituting (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 Hi to zero, we obtain an algebraic system of equations in terms of Ai, Bi, and Ci(i=0,1,2). Solving this system of algebraic equations, with the aid of Mathematica, one possible set of values of Ai, Bi, and Ci(i=0,1,2) is
(13)a=1,b=3,α=-γB22+54βk22A2+54βk32A2A2B2,λ=-54β(k22+k32)A2,A0=A2B0B2,A1=(13)A2,B1=(13)B2,C0=B0,C1=(13)B2,C2=B2,k1=-βk2(108B0k32-k22B2-k32B2+108B0k22)B2.As a result, a solution of (4a), (4b), and (4c) is(14a)u(t,x,y)=A0+A1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+A2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,(14b)v(t,x,y)=B0+B1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+B2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,(14c)w(t,x,y)=C0+C1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+C2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,where z=k1t+k2x+k3y+k4.


### 2.2. Solutions of (4a), (4b), and (4c) Using Riccati Equation as the Simplest Equation

The balancing procedure yieldsM=2 and so the solutions of (6a), (6b), and (6c) are of the form(15a)E(z)=A0+A1H+A2H2,(15b)F(z)=B0+B1H+B2H2,(15c)G(z)=C0+C1H+C2H2.Substituting (15a), (15b), and (15c) into (6a), (6b), and (6c) and using (10), we obtain an algebraic system of equations in terms of Ai, Bi, and Ci(i=0,1,2) by equating all coefficients of the functions Hi to zero. Solving the resultant system, one possible set of values is
(2.2)a=1,b=3,c=1,α=γB22-6βk22A2-6βk32A2B2A2,λ=6β(k22+k32)A2,A0=A2B0B2,A1=3A2,B1=3B2,C0=-B0,C1=-3B2,C2=-B2,k1=-βk2(-17k32B2-17k22B2+12B0k32+12B0k22)B2.Consequently, the solutions of (4a), (4b), and (4c) are(17a)u(t,x,y)=A0+A1{-b2a-θ2atanh[12θ(z+C)]}+A2{-b2a-θ2atanh[12θ(z+C)]}2,(17b)v(t,x,y)=B0+B1{-b2a-θ2atanh[12θ(z+C)]}+B2{-b2a-θ2atanh[12θ(z+C)]}2,(17c)w(t,x,y)=C0+C1{-b2a-θ2atanh[12θ(z+C)]}+C2{-b2a-θ2atanh[12θ(z+C)]}2,(18a)u(t,x,y)=A0+A1{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+A2{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,(18b)v(t,x,y)=B0+B1{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+B2{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,(18c)w(t,x,y)=C0+C1{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+C2{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,where z=k1t+k2x+k3y+k4.


## 2.1. Solutions of (4a), (4b), and (4c) Using the Bernoulli Equation as the Simplest Equation

The balancing procedure yieldsM=2. Thus, the  solutions of (6a), (6b), and (6c) are of the form(12a)E(z)=A0+A1H+A2H2,(12b)F(z)=B0+B1H+B2H2,(12c)G(z)=C0+C1H+C2H2.Substituting (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 Hi to zero, we obtain an algebraic system of equations in terms of Ai, Bi, and Ci(i=0,1,2). Solving this system of algebraic equations, with the aid of Mathematica, one possible set of values of Ai, Bi, and Ci(i=0,1,2) is
(13)a=1,b=3,α=-γB22+54βk22A2+54βk32A2A2B2,λ=-54β(k22+k32)A2,A0=A2B0B2,A1=(13)A2,B1=(13)B2,C0=B0,C1=(13)B2,C2=B2,k1=-βk2(108B0k32-k22B2-k32B2+108B0k22)B2.As a result, a solution of (4a), (4b), and (4c) is(14a)u(t,x,y)=A0+A1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+A2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,(14b)v(t,x,y)=B0+B1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+B2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,(14c)w(t,x,y)=C0+C1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+C2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,where z=k1t+k2x+k3y+k4.


## 2.2. Solutions of (4a), (4b), and (4c) Using Riccati Equation as the Simplest Equation

The balancing procedure yieldsM=2 and so the solutions of (6a), (6b), and (6c) are of the form(15a)E(z)=A0+A1H+A2H2,(15b)F(z)=B0+B1H+B2H2,(15c)G(z)=C0+C1H+C2H2.Substituting (15a), (15b), and (15c) into (6a), (6b), and (6c) and using (10), we obtain an algebraic system of equations in terms of Ai, Bi, and Ci(i=0,1,2) by equating all coefficients of the functions Hi to zero. Solving the resultant system, one possible set of values is
(2.2)a=1,b=3,c=1,α=γB22-6βk22A2-6βk32A2B2A2,λ=6β(k22+k32)A2,A0=A2B0B2,A1=3A2,B1=3B2,C0=-B0,C1=-3B2,C2=-B2,k1=-βk2(-17k32B2-17k22B2+12B0k32+12B0k22)B2.Consequently, the solutions of (4a), (4b), and (4c) are(17a)u(t,x,y)=A0+A1{-b2a-θ2atanh[12θ(z+C)]}+A2{-b2a-θ2atanh[12θ(z+C)]}2,(17b)v(t,x,y)=B0+B1{-b2a-θ2atanh[12θ(z+C)]}+B2{-b2a-θ2atanh[12θ(z+C)]}2,(17c)w(t,x,y)=C0+C1{-b2a-θ2atanh[12θ(z+C)]}+C2{-b2a-θ2atanh[12θ(z+C)]}2,(18a)u(t,x,y)=A0+A1{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+A2{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,(18b)v(t,x,y)=B0+B1{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+B2{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,(18c)w(t,x,y)=C0+C1{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+C2{sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,where z=k1t+k2x+k3y+k4.


## 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
(19)Λ1=f1(y),Λ2=f2(y),Λ3=f3(y),
and corresponding to the above multipliers we then obtain the following conserved vectors [17] of (4a), (4b), and (4c):
(20)T1t=f1(y)u,T2x=13{-3αf1(y)uv-βf1′′(y)u-3γf1(y)vw-3βf1(y)uxx+βf1′(y)uy-βf1(y)uyy},T3y=13{βf1′(y)ux-2βf1(y)uxy},T2t=f2(y)v,T2x=13{-3λf2(y)uw-βf2′′(y)v-3βf2(y)vxx+βf2′(y)vy-βf2(y)vyy},T2y=13{βf2′(y)vx-2βf2(y)vxy},T3t=f3(y)w,T3x=13{-3λf3(y)uv-βf3′′(y)w-3βf3(y)wxx+βf3′(y)wy-βf3(y)wyy},T3y=13{βf3′(y)wx-2βf3(y)wxy}.
It should be noted that due to the presence of the arbitrary function f(y) 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.



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