# Traveling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation

Authors: Wenbin Zhang; Jiangbo Zhou
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

We employ the bifurcation theory of planar dynamical system to investigate the traveling-wave solutions of the generalized Zakharov-Kuznetsov equation. Four important types of traveling wave solutions are obtained, which include the solitary wave solutions, periodic solutions, kink solutions, and antikink solutions.


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## Body

## 1. Introduction

Consider the following generalized Zakharov-Kuznetsov (ZK) equation:(1.1)ut+α(un)x+(βuxx+γuyy+δuzz)x=0,
where n≥2, α, β, γ, δ are real constants. The ZK equation was first derived for describing weakly nonlinear ion acoustic waves in a strongly magnetized lossless plasma composed of cold ions and hot isothermal electrons [1]. The ZK equation is also known as one of the two-dimensional generalizations of the KdV equation (see [2, 3]), and it is not integrable by the inverse scattering transform method [4].Whenn=2,α=(1/2)a,β=1,γ=1, and δ=1, (1.1) reduced to the equation(1.2)ut+auux+(uxx+uyy+uzz)x=0.
Wazwaz [5] obtained periodic solutions and solitary-wave solutions of (1.2) by using the sine-cosine algorithm method.Whenα=a,β=b,γ=b, and δ=0, (1.1) reduced to the equation(1.3)ut+a(un)x+b(uxx+uyy)x=0.
Wazwaz [6] obtained some solitary-wave solutions and periodic structures of (1.2) by using the extended tanh method.In this paper, we will employ the dynamical system theory [7] to investigate the traveling-wave solutions of (1.1). Numbers of smooth solitary-wave solutions, periodic solutions, kink solutions, and antikink solutions are given for each parameter condition. Here we note that such a powerful method has been employed by many authors to solve many partial differential equations [8–12].


## 2. Plane Phase Analysis

Letξ=x+y+z-ct, where c is the wave speed. By using the traveling wave transformation u(x,y,z,t)=ϕ(x+y+z-ct)=ϕ(ξ), we can reduce (1.1) to the following ordinary differential equation:(2.1)-aϕξ+b(ϕn)ξ+ϕξξξ=0,
where (·)ξ denotes the derivative of the function with respect to ξ, a=c/(β+γ+δ), and b=α/(β+γ+δ).Integrating (2.1) once and setting the integration constant as 0, we have(2.2)-aϕ+bϕn+ϕξξ=0.Letϕ′=y; then (2.2) can be transformed into the following planar dynamical system:(2.3)dϕdξ=y,dydξ=aϕ-bϕn.
We call it the traveling-wave system of (1.1). It is a planar dynamical system with Hamiltonian function(2.4)H(ϕ,y)=12y2-a2ϕ2+bn+1ϕ(n+1)=h,
where h is a constant.According to the theory of dynamical systems [7], we can obtain the properties of singular points as follows.Proposition 2.1.
Whenn=2k is even, system (2.3) has two singular points o(0,0) and A(ϕ1,0), where ϕ1=(a/b)1/(2k-1).(i)
Whena>0, o(0,0) is a saddle point and A(ϕ1,0) is a center point.(ii)
Whena=0, there is only one degenerate saddle point o(0,0).(iii)
Whena<0, o(0,0) is a center point and A(ϕ1,0) is a saddle point.Proposition 2.2.
(1) Whenn=2k+1 is odd and ab>0, system (2.3) has three singular points o(0,0) and B(±ϕ2,0), where ϕ2=(a/b)1/2k.(i)
Whena>0, o(0,0) is a saddle point and B(±ϕ2,0) are center points.(ii)
Whena=0, there is only one degenerate saddle point o(0,0).(iii)
Whena<0, o(0,0) is a center point and B(±ϕ2,0) are saddle points.
(2) Whenn=2k+1 is odd and ac≤0, system (2.3) only has one singular point o(0,0).(i)
Whena<0, o(0,0) is a saddle point or a high-order saddle point for a=0.(ii)
Whena>0, o(0,0) is a center point or a high-order center point for a=0.From the above analysis, we can obtain the bifurcations of phase portraits of system (2.3) in Figures 1 and 2.The phase portraits of system (2.3) for n=2k. (a) a>0, b>0; (b) a=0, b>0; (c) a=0, b<0; (d) a<0,  b>0; (e) a<0,  b<0; (f) a>0, b<0.
(a)(b)(c)(d)(e)(f)The phase portraits of system (2.3) for n=2k+1. (a) a>0, b>0; (b) a≤0, b>0; (c) a<0, b<0; (d) a≥0, b<0.
(a)(b)(c)(d)


## 3. Traveling Wave Solutions of (1.1)

Suppose thatϕ(ξ) is a continuous solution of (1.1) for ξ∈(-∞,+∞) and limξ→∞ϕ(ξ)=A,limξ→-∞ϕ(ξ)=B. Recall that (i) ϕ(ξ) is called a solitary wave solution if A=B and (ii) ϕ(ξ) is called a kink solutions, or antikink solution if A≠B. Usually, a solitary wave solution of (1.1) corresponds to a homoclinic orbit of its traveling wave system (2.3), a kink (or antikink) wave solution of (1.1) corresponds to a heteroclinic orbit (or the so-called connecting orbit) of system (2.3), and a periodic solution of (1.1) corresponds to a periodic orbit of system (2.3).The casen=2. As a example, we discuss the parameter region a>0, b>0 (see Figure 1(a)). In this case, system (2.4) has the form(3.1)H2(ϕ,y)=12y2-a2ϕ2+b3ϕ3=h.From Figure1(a) we can see that system (2.4) has a homoclinic orbit and a family of periodic orbits.Corresponding to the homoclinic orbit defined byH2(ϕ,y)=H2(0,0)=0, we have(3.2)y2=aϕ2-2b3ϕ3.Substituting (3.2) into the first equation of system (2.3) and integrating along the corresponding homoclinic orbit, we obtain a smooth solitary wave solution:(3.3)u1(ξ)=3a2bsech2(a2ξ).Corresponding to the family of periodic orbits defined byH2(ϕ,y)=h, h(h1,0), we have(3.4)y2=2b3(ϕ-r1)(ϕ-r2)(r3-ϕ),
where r1, r2, r3 are three real roots of the equation aϕ2-(2b/3)ϕ3+2h=0 and h1=H2(ϕ1,0)=-a3/6b2. Thus, we obtain a periodic solution:(3.5)u2(ξ)=r3-(r3-r2)sn2(b(r3-r1)6ξ,r3-r2r3-r1).The casen=3. In this case, system (2.4) has the form(3.6)H3(ϕ,y)=12y2-a2ϕ2+b4ϕ4=h.(1)
From Figure2(a) we can see that system (2.4) has two homoclinic orbits and three families of periodic orbits.Corresponding to the two homoclinic orbits defined byH3(ϕ,y)=H3(0,0)=0, we have(3.7)y2=aϕ2-b2ϕ4.Substituting (3.7) into the first equation of system (2.3), and integrating along the corresponding homoclinic orbits, we obtain two smooth solitary wave solutions:(3.8)u3,4(ξ)=±2absech(aξ).Corresponding to the two families of periodic orbits defined byH3(ϕ,y)=h, h(h2,0), we have(3.9)y2=aϕ2-b2ϕ4+2h,
where h2=H3(±ϕ2,0)=-a2/4b.Substituting (3.9) into the first equation of system (2.3) and integrating along the corresponding periodic orbit, we obtain two periodic solutions:(3.10)u5,6(ξ)=±a+kbdn(a+k2ξ,2ka+k),
where k=a2+4bh.Corresponding to the family of periodic orbits defined byH3(ϕ,y)=h, h∈(0,∞), we have(3.11)y2=aϕ2-b2ϕ4+2h.
Substituting (3.11) into the first equation of system (2.3) and integrating along the corresponding periodic orbit, we obtain a periodic solution:(3.12)u7(ξ)=a+kbdn(kξ,a+k2k).(2)
From Figure2(c) we can see that system (2.4) has two heteroclinic orbits and a family of periodic orbits.Corresponding to the two heteroclinic orbits defined byH3(ϕ,y)=h2, we have(3.13)y2=aϕ2-b2ϕ4-a22b.Substituting (3.13) into the first equation of system (2.3) and integrating along the corresponding heteroclinic orbits, we obtain kink solutions, and antikink solutions:(3.14)u8,9(ξ)=±abtanh(-a2ξ).Corresponding to the family of periodic orbits defined byH3(ϕ,y)=h, h∈(0,h2), we have(3.15)y2=aϕ2-b2ϕ4+2h.Substituting (3.15) into the first equation of system (2.3) and integrating along the corresponding periodic orbits, we obtain a periodic solution:(3.16)u10(ξ)=a+kbsn(-aξ,a-k2a).(3)
From Figure2(b) we can see that system (2.4) has a family of periodic orbits.Corresponding to the family of periodic orbits defined byH3(ϕ,y)=h, h∈(0,∞), we have the same periodic solution of u(ξ) as (3.12).Specifically, whenc=0, (3.12) has the form(3.17)y2=-b2ϕ4+2h,h∈(0,∞).Substituting (3.17) into the first equation of system (2.3) and integrating along the corresponding periodic orbits, we obtain a periodic solution:(3.18)u11(ξ)=2hbcn(2hξ,22).The casen>3.(1)
Whenn=2k is even, from Figure 1(a) we can see that system (2.4) has a homoclinic orbit.Corresponding to the homoclinic orbit defined byH(ϕ,y)=H(0,0)=0, we have(3.19)y2=aϕ2-2b(2k+1)ϕ2k+1.
Substituting (3.19) into the first equation of system (2.3), we have(3.20)∫0ϕ1sa-(2b/(2k+1))s(2k-1)ds=±∫0ξds.
Let(3.21)φ=s2k-1,φ1=ϕ2k-1.
Thus, (3.20) and (3.21) merge into(3.22)∫0φ11φa-(2b/(2k+1))φdφ=(2k-1)|ξ|.Completing the integral in (3.22), we obtain(3.23)φ(ξ)=(2k+1)a2bsech2((2k-1)a2ξ).From (3.21) and (3.23), we have(3.24)u12(ξ)=[(2k+1)a2bsech2((2k-1)a2ξ)]1/(2k-1).(2)
Whenn=2k+1 is odd, from Figure 2(a) we can see that system (2.4) has two homoclinic orbits. Corresponding to the homoclinic orbits defined by H(ϕ,y)=H(0,0)=0, we have(3.25)y2=aϕ2-b(k+1)ϕ2k+2.
Substituting (3.25) into the first equation of system (2.3), we have(3.26)∫0ϕ1s(A-sk)(A+sk)ds=±b(k+1)∫0ξds,
where A=a(k+1)/b.Let(3.27)ψ=sk,ψ1=ϕk.
Thus, (3.26) and (3.27) merge into(3.28)∫0ψ11ψ(A-ψ)(A+ψ)dψ=kb(k+1)|ξ|.
Completing the integral in (3.28), we obtain(3.29)ψ(ξ)=±(k+1)absech(akξ).
From (3.27) and (3.29), we have(3.30)u13,14(ξ)=±[(k+1)absech(akξ)]1/k.



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*Source: 107846-2012-03-13.xml*