Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 747295, 14 pages

doi:10.1155/2012/747295

Research Article

Traveling Wave Solutions for the Generalized Zakharov Equations

Ming Song1,2 and Zhengrong Liu¹

¹Department of Mathematics, South China University of Technology, Guangzhou 510640, China

²Department of Mathematics, Faculty of Sciences, Yuxi Normal University, Yuxi 653100, China

Received 12 March 2012; Accepted 30 March 2012

Academic Editor: Anders Eriksson

Copyright © 2012 Ming Song and Zhengrong Liu. 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 use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

1. Introduction

The Zakharov equations

which is one of the fundamental models governing dynamics of nonlinear waves in one-dimensional systems. It describes the interaction between high-frequency and low-frequency waves. The physically most important example involves the interaction between the Langmuir and ion-acoustic waves in plasmas [1]. The equations can be derived from a hydrodynamic description of the plasma [2, 3]. However, some important effects such as transit-time damping and ion nonlinearities, which are also implied by the fact that the values used for the ion damping have been anomalously large from the point of view of linear ion-acoustic wave dynamics, have been ignored in (1.1). This is equivalent to saying that (1.1) is a simplified model of strong Langmuir turbulence. Thus we have to generalize (1.1) by taking more elements into account. Starting from the dynamical plasma equations with the help of relaxed Zakharov simplification assumptions, and through making use of the time-averaged two-time-scale two-fluid plasma description, (1.1) are generalized to contain the self-generated magnetic field [4, 5], and the first related study on magnetized plasmas in [6, 7]. The generalized Zakharov equations are a set of coupled equations and may be written as [8]

Malomed et al. [8] analyzed internal vibrations of a solitary wave in (1.2) by means of a variational approach. Wang and Li [9] obtained a number of periodic wave solutions of (1.2) by using extended F-expansion method. Javidi and Golbabai [10] used the He's variational iteration method to obtain solitary wave solutions of (1.2). Zhang [11] obtained the exact traveling wave solutions of (1.2) by using the direct algebraic method. Zhang [12] used He’s semi-inverse method to search for solitary wave solutions of (1.2). Javidi and Golbabai [13] obtained the exact and numerical solutions of (1.2) by using the variational iteration method. Li et al. [14] used the Exp-function method to seek exact solutions of (1.2). Borhanifar et al. [15] obtained the generalized solitary solutions and periodic solutions of (1.2) by using the Exp-function method. Khan et al. [16] used He's variational approach to obtain new soliton solutions of (1.2).

The aim of this paper is to study the traveling wave solutions and their limits for (1.2) by using the bifurcation method and qualitative theory of dynamical systems [17–24]. Through some special phase orbits, we obtain many smooth periodic wave solutions and periodic blow-up solutions. Their limits contain kink-profile solitary wave solutions, unbounded wave solutions, periodic blow-up solutions, and solitary wave solutions.

The remainder of this paper is organized as follows. In Section 2, by using the bifurcation theory of planar dynamical systems, two-phase portraits for the corresponding traveling wave system of (1.2) are given under different parameter conditions. The relations between the traveling wave solutions and the Hamiltonian are presented. In Section 3, we obtain a number of traveling wave solutions of (1.2) and give the relations of the traveling wave solutions. A short conclusion will be given in Section 4.

2. Phase Portraits and Qualitative Analysis

We assume that the traveling wave solutions of (1.2) is of the form

where and are real functions; , and are real constants.

Substituting (2.1) into (1.2), we have

Integrating the second equation of (2.2) twice, and letting the first integral constant be zero, we have

where is integral constant.

Substituting (2.3) into the first equation of (2.2), we have

Letting , , and , then we get the following planar system

Obviously, the above system (2.5) is a Hamiltonian system with Hamiltonian function

In order to investigate the phase portrait of (2.5), set

Obviously, has three zero points, , , and , which are given as follows:

Letting be one of the singular points of system (2.5), then the characteristic values of the linearized system of system (2.5) at the singular points are

From the qualitative theory of dynamical systems, we know that:

(1)if , is a saddle point;(2)if , is a center point;(3)if , is a degenerate saddle point;

Therefore, we obtain the phase portraits of system (2.5) in Figure 1.

(a)

(b)

Figure 1: The phase portraits of system (2.5). (a) , (b) .

Let

where is Hamiltonian.

Next, we consider the relations between the orbits of (2.5) and the Hamiltonian .

Set

According to Figure 1, we get the following propositions.

Proposition 2.1. Suppose that and , we have the following.

(1)When or , system (2.5) does not have any closed orbit.(2)When , system (2.5) has three periodic orbits , , and .(3)When , system (2.5) has two periodic orbits and .(4)When , system (2.5) has two heteroclinic orbits and .

Proposition 2.2. Suppose that and , we have the following.

(1)When , system (2.5) does not have any closed orbit.(2)When , system (2.5) has two periodic orbits and .(3)When , system (2.5) has two homoclinic orbits and .(4)When , system (2.5) has a periodic orbit .

From the qualitative theory of dynamical systems, we know that a smooth solitary wave solution of a partial differential system corresponds to a smooth homoclinic orbit of a traveling wave equation. A smooth kink wave solution or an unbounded wave solution corresponds to a smooth heteroclinic orbit of a traveling wave equation. Similarly, a periodic orbit of a traveling wave equation corresponds to a periodic traveling wave solution of a partial differential system. According to the above analysis, we have the following propositions.

Proposition 2.3. If and , we have the following.

(1)When , (1.2) has two periodic wave solutions (corresponding to the periodic orbit in Figure 1) and two periodic blow-up wave solutions (corresponding to the periodic orbits and in Figure 1).(2)When , (1.2) has two periodic blow-up wave solutions (corresponding to the periodic orbits and in Figure 1).(3)When , (1.2) has two kink-profile solitary wave solutions and two unbounded wave solutions (corresponding to the heteroclinic orbits and in Figure 1).

Proposition 2.4. If and , we have the following.

(1)When , (1.2) has two periodic wave solutions (corresponding to the periodic orbits and in Figure 1).(2)When , (1.2) has two solitary wave solutions (corresponding to the homoclinic orbits and in Figure 1).(3)When , (1.2) has two periodic wave solutions (corresponding to the periodic orbit in Figure 1).

3. Traveling Wave Solutions and Their Relations

Firstly, we will obtain the explicit expressions of traveling wave solutions for the (1.2) when and .

From the phase portrait, we note that there are three periodic orbits , , and passing the points , and . In plane the expressions of the orbits are given as

where , , , , and .

Substituting (3.1) into and integrating them along , and , we have

Completing above integrals we obtain

Noting that (2.1) and (2.3), we get the following periodic wave solutions:

where and .

From the phase portrait, we note that there are two special orbits and , which have the same hamiltonian as that of the center point . In plane the expressions of the orbits are given as

where and .

Substituting (3.5) into , and integrating them along the two orbits and , it follows that

Completing above integrals we obtain

Noting (2.1) and (2.3), we get the following periodic blow-up wave solutions:

where and .

From the phase portrait, we see that there are two heteroclinic orbits and connected at saddle points and . In plane the expressions of the heteroclinic orbits are given as

Substituting (3.9) into , and integrating them along the heteroclinic orbits and , it follows that

Completing above integrals we obtain

Noting (2.1) and (2.3), we get the following kink profile solitary wave solutions:

and unbounded wave solutions

where and .

Secondly, we will obtain the explicit expressions of traveling wave solutions for (1.2) when and .

From the phase portrait, we see that there are two closed orbits and passing the points , and . In plane the expressions of the closed orbits are given as

where , , , , and .

Substituting (3.14) into , and integrating them along and , we have

Completing above integrals we obtain

where and .

Noting (2.1) and (2.3), we get the following periodic wave solutions:

where and .

From the phase portrait, we see that there are two symmetric homoclinic orbits and connected at the saddle point . In plane the expressions of the homoclinic orbits are given as

where and .

Substituting (3.18) into , and integrating them along the orbits and , we have

Completing above integrals we obtain

Noting (2.1) and (2.3), we get the following solitary wave solutions:

where and .

From the phase portrait, we see that there is a closed orbit passing the points and . In plane the expressions of the closed orbits are given as

where , , , , and .

Substituting (3.22) into , and integrating them along the orbit , we have

Completing above integrals we obtain

Noting (2.1) and (2.3), we get the following periodic wave solutions:

where and .

Thirdly, we will give the relations of the traveling wave solutions.

(1)Letting , it follows that , , and . Therefore, we obtain , , and .(2)Letting , it follows that , , and . Therefore, we obtain and .(3)Letting , it follows that , , , , , and . Therefore, we obtain and .(4)Letting , it follows that , , , , and . Therefore, we obtain and .(5)Letting , it follows that , , , and . Therefore, we obtain , , and .

Finally, we will show that the periodic wave solutions evolute into the kink-profile solitary wave solutions when the Hamiltonian (corresponding to the changes of phase orbits of Figure 1 as varies). We take some suitable choices of the parameters, such as

as an illustrative sample and draw their plots (see Figures 2 and 3).

(a)

(b)

(c)

Figure 2: The real part of the periodic wave solution evolutes into the kink-profile solitary wave solutions with the conditions (3.26). (a) ; (b) ; (c) .

(a)

(b)

(c)

Figure 3: The imaginary part of the periodic wave solution evolutes into the kink-profile solitary wave solutions with the conditions (3.26). (a) ; (b) ; (c) .

4. Conclusion

In this paper, we obtain phase portraits for the corresponding traveling wave system of (1.2) by using the bifurcation theory of planar dynamical systems. Furthermore, a number of exact traveling wave solutions are also obtained, and their relations are given. The method can be applied to many other nonlinear evolution equations, and we believe that many new results wait for further discovery by this method.

Acknowledgment

Research is supported by the National Natural Science Foundation of China (Grant no. 11171115) and the Natural Science Foundation of Yunnan Province (Grant no. 2010ZC154).

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