Traveling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation

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.

1. Introduction

Consider the following generalized Zakharov-Kuznetsov (ZK) equation:𝑢_𝑡+𝛼(𝑢^𝑛)_𝑥+𝛽𝑢_𝑥𝑥+𝛾𝑢_𝑦𝑦+𝛿𝑢_𝑧𝑧_𝑥=0,(1.1) 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].

When 𝑛=2,𝛼=(1/2)𝑎,𝛽=1,𝛾=1, and 𝛿=1, (1.1) reduced to the equation𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧_𝑥=0.(1.2) Wazwaz [5] obtained periodic solutions and solitary-wave solutions of (1.2) by using the sine-cosine algorithm method.

When 𝛼=𝑎,𝛽=𝑏,𝛾=𝑏, and 𝛿=0, (1.1) reduced to the equation𝑢_𝑡+𝑎(𝑢^𝑛)_𝑥+𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦_𝑥=0.(1.3) 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

Let 𝜉=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:−𝑎𝜙_𝜉+𝑏(𝜙^𝑛)_𝜉+𝜙_𝜉𝜉𝜉=0,(2.1) 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−𝑎𝜙+𝑏𝜙^𝑛+𝜙_𝜉𝜉=0.(2.2)

Let 𝜙′=𝑦; then (2.2) can be transformed into the following planar dynamical system:𝑑𝜙𝑑𝑦𝑑𝜉=𝑦,𝑑𝜉=𝑎𝜙−𝑏𝜙^𝑛.(2.3) We call it the traveling-wave system of (1.1). It is a planar dynamical system with Hamiltonian function𝐻(𝜙,𝑦)=12𝑦²−𝑎2𝜙²+𝑏𝑛+1𝜙^(𝑛+1)=ℎ,(2.4) 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. When 𝑛=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)When 𝑎>0, 𝑜(0,0) is a saddle point and 𝐴(𝜙₁,0) is a center point.(ii)When 𝑎=0, there is only one degenerate saddle point 𝑜(0,0).(iii)When 𝑎<0, 𝑜(0,0) is a center point and 𝐴(𝜙₁,0) is a saddle point.

Proposition 2.2. (1) When 𝑛=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)When 𝑎>0, 𝑜(0,0) is a saddle point and 𝐵(±𝜙₂,0) are center points.(ii)When 𝑎=0, there is only one degenerate saddle point 𝑜(0,0).(iii)When 𝑎<0, 𝑜(0,0) is a center point and 𝐵(±𝜙₂,0) are saddle points.
(2) When 𝑛=2𝑘+1 is odd and 𝑎𝑐≤0, system (2.3) only has one singular point 𝑜(0,0).(i)When 𝑎<0, 𝑜(0,0) is a saddle point or a high-order saddle point for 𝑎=0.(ii)When 𝑎>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.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

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)

Figure 2 

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.

3. Traveling Wave Solutions of (1.1)

Suppose that 𝜙(𝜉) is a continuous solution of (1.1) for 𝜉∈(−∞,+∞) and lim_𝜉→∞𝜙(𝜉)=𝐴,lim_{𝜉→−∞}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 case 𝑛=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 form𝐻₂(𝜙,𝑦)=12𝑦²−𝑎2𝜙²+𝑏3𝜙³=ℎ.(3.1)

From Figure 1(a) we can see that system (2.4) has a homoclinic orbit and a family of periodic orbits.

Corresponding to the homoclinic orbit defined by 𝐻₂(𝜙,𝑦)=𝐻₂(0,0)=0, we have𝑦²=𝑎𝜙²−2𝑏3𝜙³.(3.2)

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:𝑢₁(𝜉)=3𝑎2𝑏secℎ²√𝑎2𝜉.(3.3)

Corresponding to the family of periodic orbits defined by 𝐻₂(𝜙,𝑦)=ℎ, ℎ(ℎ₁,0), we have𝑦²=2𝑏3𝜙−𝑟¹𝜙−𝑟₂𝑟₃−𝜙,(3.4) 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:𝑢₂(𝜉)=𝑟₃−𝑟₃−𝑟₂𝑠𝑛²⎛⎜⎜⎝𝑏𝑟₃−𝑟₁6𝜉,𝑟₃−𝑟₂𝑟₃−𝑟₁⎞⎟⎟⎠.(3.5)The case 𝑛=3. In this case, system (2.4) has the form𝐻₃(𝜙,𝑦)=12𝑦²−𝑎2𝜙²+𝑏4𝜙⁴=ℎ.(3.6)(1)From Figure 2(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 by 𝐻₃(𝜙,𝑦)=𝐻₃(0,0)=0, we have𝑦²=𝑎𝜙²−𝑏2𝜙⁴.(3.7)

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:𝑢_3,4(𝜉)=±2𝑎𝑏secℎ√𝑎𝜉.(3.8)

Corresponding to the two families of periodic orbits defined by 𝐻₃(𝜙,𝑦)=ℎ, ℎ(ℎ₂,0), we have𝑦²=𝑎𝜙²−𝑏2𝜙⁴+2ℎ,(3.9) 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:𝑢_5,6(𝜉)=±𝑎+𝑘𝑏𝑑𝑛𝑎+𝑘2𝜉,2𝑘𝑎+𝑘,(3.10) where 𝑘=√𝑎²+4𝑏ℎ.

Corresponding to the family of periodic orbits defined by 𝐻₃(𝜙,𝑦)=ℎ, ℎ∈(0,∞), we have𝑦²=𝑎𝜙²−𝑏2𝜙⁴+2ℎ.(3.11) Substituting (3.11) into the first equation of system (2.3) and integrating along the corresponding periodic orbit, we obtain a periodic solution:𝑢₇(𝜉)=𝑎+𝑘𝑏𝑑𝑛√𝑘𝜉,𝑎+𝑘2𝑘.(3.12)(2)From Figure 2(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 by 𝐻₃(𝜙,𝑦)=ℎ₂, we have𝑦²=𝑎𝜙²−𝑏2𝜙⁴−𝑎²2𝑏.(3.13)

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:𝑢_8,9(𝜉)=±𝑎𝑏tanℎ−𝑎2𝜉.(3.14)

Corresponding to the family of periodic orbits defined by 𝐻₃(𝜙,𝑦)=ℎ, ℎ∈(0,ℎ₂), we have𝑦²=𝑎𝜙²−𝑏2𝜙⁴+2ℎ.(3.15)

Substituting (3.15) into the first equation of system (2.3) and integrating along the corresponding periodic orbits, we obtain a periodic solution:𝑢₁₀(𝜉)=𝑎+𝑘𝑏𝑠𝑛√−𝑎𝜉,𝑎−𝑘2𝑎.(3.16)(3) From Figure 2(b) we can see that system (2.4) has a family of periodic orbits.

Corresponding to the family of periodic orbits defined by 𝐻₃(𝜙,𝑦)=ℎ, ℎ∈(0,∞), we have the same periodic solution of 𝑢(𝜉) as (3.12).

Specifically, when 𝑐=0, (3.12) has the form𝑦²=−𝑏2𝜙⁴+2ℎ,ℎ∈(0,∞).(3.17)

Substituting (3.17) into the first equation of system (2.3) and integrating along the corresponding periodic orbits, we obtain a periodic solution:𝑢₁₁(𝜉)=2ℎ𝑏𝑐𝑛2√ℎ𝜉,√22.(3.18)

The case 𝑛>3.(1)When 𝑛=2𝑘 is even, from Figure 1(a) we can see that system (2.4) has a homoclinic orbit.

Corresponding to the homoclinic orbit defined by 𝐻(𝜙,𝑦)=𝐻(0,0)=0, we have𝑦²=𝑎𝜙²−2𝑏(2𝑘+1)𝜙^2𝑘+1.(3.19) Substituting (3.19) into the first equation of system (2.3), we have^𝜙01𝑠√𝑎−(2𝑏/(2𝑘+1))𝑠^(2𝑘−1)𝑑𝑠=±^𝜉0𝑑𝑠.(3.20) Let𝜑=𝑠^2𝑘−1,𝜑₁=𝜙^2𝑘−1.(3.21) Thus, (3.20) and (3.21) merge into^𝜑1₀1𝜑√𝑎−(2𝑏/(2𝑘+1))𝜑𝑑𝜑=(2𝑘−1)||𝜉||.(3.22)

Completing the integral in (3.22), we obtain𝜑(𝜉)=(2𝑘+1)𝑎2𝑏secℎ²(2𝑘−1)√𝑎2𝜉.(3.23)

From (3.21) and (3.23), we have𝑢₁₂(𝜉)=(2𝑘+1)𝑎2𝑏secℎ²(2𝑘−1)√𝑎2𝜉^{1/(2𝑘−1)}.(3.24)(2) When 𝑛=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 have𝑦²=𝑎𝜙²−𝑏(𝑘+1)𝜙^2𝑘+2.(3.25) Substituting (3.25) into the first equation of system (2.3), we have^𝜙01𝑠𝐴−𝑠^𝑘𝐴+𝑠^𝑘𝑑𝑠=±𝑏(𝑘+1)^𝜉0𝑑𝑠,(3.26) where 𝐴=√𝑎(𝑘+1)/𝑏.

Let𝜓=𝑠^𝑘,𝜓₁=𝜙^𝑘.(3.27) Thus, (3.26) and (3.27) merge into^𝜓1₀1𝜓√(𝐴−𝜓)(𝐴+𝜓)𝑑𝜓=𝑘𝑏(𝑘+1)||𝜉||.(3.28) Completing the integral in (3.28), we obtain𝜓(𝜉)=±(𝑘+1)𝑎𝑏secℎ√𝑎𝑘𝜉.(3.29) From (3.27) and (3.29), we have𝑢_13,14(𝜉)=±(𝑘+1)𝑎𝑏secℎ√𝑎𝑘𝜉^1/𝑘.(3.30)

Acknowledgments

This work was supported by the Startup Fund for Advanced Talents of Jiangsu University (no. 09JDG013), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), the Jiangsu Planned Projects for Postdoctoral Research Funds (no. 0902107C), the Jiangsu Government Scholarship for Overseas Studies, the Taizhou Social Development Project (no. 2011213), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.