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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 107846, 10 pages
http://dx.doi.org/10.5402/2012/107846
Research Article

Traveling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation

1Taizhou Institute of Science and Technology, NanJing University of Science and Technology, Taizhou 225300, China
2Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Received 19 December 2011; Accepted 4 January 2012

Academic Editors: G. Akrivis and M. Winter

Copyright © 2012 Wenbin Zhang and Jiangbo Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:𝑢𝑡+𝛼(𝑢𝑛)𝑥+𝛽𝑢𝑥𝑥+𝛾𝑢𝑦𝑦+𝛿𝑢𝑧𝑧𝑥=0,(1.1) where 𝑛2, 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 equation𝑢𝑡+𝑎𝑢𝑢𝑥+𝑢𝑥𝑥+𝑢𝑦𝑦+𝑢𝑧𝑧𝑥=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 equation𝑢𝑡+𝑎(𝑢𝑛)𝑥𝑢+𝑏𝑥𝑥+𝑢𝑦𝑦𝑥=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 [812].

2. Plane Phase Analysis

Let 𝜉=𝑥+𝑦+𝑧𝑐𝑡, where 𝑐 is the wave speed. By using the traveling wave transformation 𝑢(𝑥,𝑦,𝑧,𝑡)=𝜙(𝑥+𝑦+𝑧𝑐𝑡)=𝜙(𝜉), we can reduce (1.1) to the following ordinary differential equation:𝑎𝜙𝜉+𝑏(𝜙𝑛)𝜉+𝜙𝜉𝜉𝜉=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 have𝑎𝜙+𝑏𝜙𝑛+𝜙𝜉𝜉=0.(2.2)

Let 𝜙=𝑦; then (2.2) can be transformed into the following planar dynamical system:𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦𝑑𝜉=𝑎𝜙𝑏𝜙𝑛.(2.3) We call it the traveling-wave system of (1.1). It is a planar dynamical system with Hamiltonian function1𝐻(𝜙,𝑦)=2𝑦2𝑎2𝜙2+𝑏𝜙𝑛+1(𝑛+1)=,(2.4) where 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/(2𝑘1).(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 𝑎𝑐0, 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.

fig1
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.
fig2
Figure 2: The phase portraits of system (2.3) for 𝑛=2𝑘+1. (a) 𝑎>0, 𝑏>0; (b) 𝑎0, 𝑏>0; (c) 𝑎<0, 𝑏<0; (d) 𝑎0, 𝑏<0.

3. Traveling Wave Solutions of (1.1)

Suppose that 𝜙(𝜉) is a continuous solution of (1.1) for 𝜉(,+) and lim𝜉𝜙(𝜉)=𝐴,lim𝜉𝜙(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 𝐴𝐵. 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 form𝐻21(𝜙,𝑦)=2𝑦2𝑎2𝜙2+𝑏3𝜙3=.(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=𝑎𝜙22𝑏3𝜙3.(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)=3𝑎2𝑏sec2𝑎2𝜉.(3.3)

Corresponding to the family of periodic orbits defined by 𝐻2(𝜙,𝑦)=, (1,0), we have𝑦2=2𝑏3𝜙𝑟1𝜙𝑟2𝑟3,𝜙(3.4) where 𝑟1, 𝑟2, 𝑟3 are three real roots of the equation 𝑎𝜙2(2𝑏/3)𝜙3+2=0 and 1=𝐻2(𝜙1,0)=𝑎3/6𝑏2. Thus, we obtain a periodic solution:𝑢2(𝜉)=𝑟3𝑟3𝑟2𝑠𝑛2𝑏𝑟3𝑟16𝜉,𝑟3𝑟2𝑟3𝑟1.(3.5)The case 𝑛=3. In this case, system (2.4) has the form𝐻31(𝜙,𝑦)=2𝑦2𝑎2𝜙2+𝑏4𝜙4=.(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=𝑎𝜙2𝑏2𝜙4.(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:𝑢3,4(𝜉)=±2𝑎𝑏sec.𝑎𝜉(3.8)

Corresponding to the two families of periodic orbits defined by 𝐻3(𝜙,𝑦)=, (2,0), we have𝑦2=𝑎𝜙2𝑏2𝜙4+2,(3.9) where 2=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:𝑢5,6(𝜉)=±𝑎+𝑘𝑏𝑑𝑛𝑎+𝑘2𝜉,2𝑘𝑎+𝑘,(3.10) where 𝑘=𝑎2+4𝑏.

Corresponding to the family of periodic orbits defined by 𝐻3(𝜙,𝑦)=, (0,), we have𝑦2=𝑎𝜙2𝑏2𝜙4+2.(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+𝑘𝑏𝑑𝑛𝑘𝜉,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)=2, we have𝑦2=𝑎𝜙2𝑏2𝜙4𝑎2.2𝑏(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:𝑢8,9(𝜉)=±𝑎𝑏tan𝑎2𝜉.(3.14)

Corresponding to the family of periodic orbits defined by 𝐻3(𝜙,𝑦)=, (0,2), we have𝑦2=𝑎𝜙2𝑏2𝜙4+2.(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+𝑘𝑏𝑠𝑛𝑎𝜉,𝑎𝑘2𝑎.(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)=, (0,), we have the same periodic solution of 𝑢(𝜉) as (3.12).

Specifically, when 𝑐=0, (3.12) has the form𝑦2𝑏=2𝜙4+2,(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)=2𝑏2𝑐𝑛𝜉,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=𝑎𝜙22𝑏𝜙(2𝑘+1)2𝑘+1.(3.19) Substituting (3.19) into the first equation of system (2.3), we have𝜙01𝑠𝑎(2𝑏/(2𝑘+1))𝑠(2𝑘1)𝑑𝑠=±𝜉0𝑑𝑠.(3.20) Let𝜑=𝑠2𝑘1,𝜑1=𝜙2𝑘1.(3.21) Thus, (3.20) and (3.21) merge into𝜑101𝜑||𝜉||.𝑎(2𝑏/(2𝑘+1))𝜑𝑑𝜑=(2𝑘1)(3.22)

Completing the integral in (3.22), we obtain𝜑(𝜉)=(2𝑘+1)𝑎2𝑏sec2(2𝑘1)𝑎2𝜉.(3.23)

From (3.21) and (3.23), we have𝑢12(𝜉)=(2𝑘+1)𝑎2𝑏sec2(2𝑘1)𝑎2𝜉1/(2𝑘1).(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=𝑎𝜙2𝑏𝜙(𝑘+1)2𝑘+2.(3.25) Substituting (3.25) into the first equation of system (2.3), we have𝜙01𝑠𝐴𝑠𝑘𝐴+𝑠𝑘𝑑𝑠=±𝑏(𝑘+1)𝜉0𝑑𝑠,(3.26) where 𝐴=𝑎(𝑘+1)/𝑏.

Let𝜓=𝑠𝑘,𝜓1=𝜙𝑘.(3.27) Thus, (3.26) and (3.27) merge into𝜓101𝜓(𝐴𝜓)(𝐴+𝜓)𝑑𝜓=𝑘𝑏||𝜉||.(𝑘+1)(3.28) Completing the integral in (3.28), we obtain𝜓(𝜉)=±(𝑘+1)𝑎𝑏sec.𝑎𝑘𝜉(3.29) From (3.27) and (3.29), we have𝑢13,14(𝜉)=±(𝑘+1)𝑎𝑏sec𝑎𝑘𝜉1/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.

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