Abstract and Applied Analysis

Volume 2012 (2012), Article ID 308326, 8 pages

doi:10.1155/2012/308326

Research Article

Application of Bifurcation Method to the Generalized Zakharov Equations

Ming Song

Department of Mathematics, Yuxi Normal University, Yuxi 653100, China

Received 22 September 2012; Revised 12 October 2012; Accepted 12 October 2012

Academic Editor: Irena Lasiecka

Copyright © 2012 Ming Song. 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 blow-up wave solutions and solitary wave solutions.

1. Introduction

The Zakharov equations

are of the fundamental models governing dynamics of nonlinear waves in one-dimensional systems. The Zakharov equation describe 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 [4–6] and the first related study on magnetized plasmas in [7, 8]. The generalized Zakharov equations are a set of coupled equations and may be written as [9]

Malomed et al. [9] analyzed internal vibrations of a solitary wave in (1.2) by means of a variational approach. Wang and Li [10] obtained a number of periodic wave solutions of (1.2) by using extended F-expansion method. Javidi and Golbabai [11] used the He’s variational iteration method to obtain solitary wave solutions of (1.2). Layeni [12] obtained the exact traveling wave solutions of (1.2) by using the new rational auxiliary equation method. Zhang [13] used He’s semi-inverse method to search for solitary wave solutions of (1.2). Javidi and Golbabai [14] obtained the exact and numerical solutions of (1.2) by using the variational iteration method. Li et al. [15] used the Exp-function method to seek exact solutions of (1.2). Borhanifar et al. [16] obtained the generalized solitary solutions and periodic solutions of (1.2) by using the Exp-function method. Khan et al. [17] used He’s variational approach to obtain new soliton solutions of (1.2). Song and Liu [18] obtained a number of traveling wave solutions of (1.2) by using bifurcation method of dynamical systems.

In this work, we aim to study new generalized Zakharov equations.

where is positive integer. First, we aim to apply the bifurcation method of dynamical systems [18–22] to study the phase portraits for the corresponding traveling wave system of (1.3). Our second aim is to obtain exact traveling wave solutions for (1.3).

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 (1.3) 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 (1.3). A short conclusion will be given in Section 4.

2. Phase Portraits and Qualitative Analysis

We assume that the traveling wave solutions of (1.3) is of the form

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

Substituting (2.1) into (1.3), we have

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

where is the second integral constant.

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

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

Obviously, 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 bifurcation phase portraits of system (2.5) in Figure 1.

Figure 1: The bifurcation phase portraits of system (2.5). (a) , , (b) , , (c) , , (d) , and .

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 , one has:

(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 , one has:

(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 .

Proposition 2.3. (1) When , , and , system (2.5) has a periodic orbits.

(2) When , , system (2.5) does not have any closed 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 a 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 2.4. If and , one has the following:

(1)when , (1.3) 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 , (1.3) has periodic blow-up wave solutions (corresponding to the periodic orbits and in Figure 1);(3)when , (1.3) has two kink profile solitary wave solutions and two unbounded wave solutions (corresponding to the heteroclinic orbits and in Figure 1).

Proposition 2.5. If and , one has the following:

(1)when , (1.3) has two periodic wave solutions (corresponding to the periodic orbits and in Figure 1);(2)when , (1.3) has two solitary wave solutions (corresponding to the homoclinic orbits and in Figure 1);(3)when , (1.3) has two periodic wave solutions (corresponding to the periodic orbit in Figure 1).

3. Exact Traveling Wave Solutions of (1.3)

Firstly, we will obtain the explicit expressions of traveling wave solutions for (1.3) when and . From the phase portrait, we note that there are two special orbits and , which have the same Hamiltonian with that of the center point . In plane, the expressions of the orbits are given as

Substituting (3.1) into and integrating them along the two orbits and , it follows that

where .

Completing above integrals, we obtain

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

where and .

Secondly, we will obtain the explicit expressions of traveling wave solutions for (1.3) when 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

Substituting (3.5) 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 .

4. Conclusion

In this paper, we obtain phase portraits for the corresponding traveling wave system of (1.3) by using the bifurcation theory of planar dynamical systems. Furthermore, a number of exact traveling wave solutions are also obtained. 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.

Acknowledgments

The author would like to thank the anonymous referees for their useful and valuable suggestions. This work is supported by the Natural Science Foundation of Yunnan Province (no. 2010ZC154).

References

V. E. Zakharov, “Collapse of Langmuir waves,” Soviet Physics, JETP, vol. 35, pp. 908–914, 1972.

M. V. Goldman, “Langmuir wave solitons and spatial collapse in plasma physics,” Physica D, vol. 18, no. 1–3, pp. 67–76, 1986.

D. R. Nicolson, Introduction to Plasma Theory, John Wiley & Sons, New York, NY, USA, 1983.

N. L. Tsintsadze, D. D. Tskhakaya, and L. Stenflo, “Modulational instabilities due to relativistic electron mass variations,” Physics Letters A, vol. 72, no. 2, pp. 115–116, 1979.

K. B. Dysthe, E. Mjølhus, H. L. Pécseli, and L. Stenflo, “Langmuir solitons in magnetized plasmas,” Plasma Physics, vol. 20, no. 11, pp. 1087–1099, 1978.

L. H. Li, “Langmuir turbulence equations with the self-generated magnetic field,” Physics of Fluids B, vol. 5, no. 2, pp. 350–356, 1993.

A. N. Kaufman and L. Stenflo, “Upper-hybrid solitons,” Physica Scripta, vol. 11, pp. 269–270, 1975.

R. T. Faria Jr. and P. K. Shukla, “Generalized Kaufman-Stenflo equations for collisional magnetoplasmas,” Physica Scripta, vol. T82, pp. 7–9, 1999.

B. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Dynamics of solitary waves in the Zakharov model equations,” Physical Review E, vol. 55, no. 1, pp. 962–968, 1997.

M. Wang and X. Li, “Extended -expansion method and periodic wave solutions for the generalized Zakharov equations,” Physics Letters A, vol. 343, no. 1–3, pp. 48–54, 2005.

M. Javidi and A. Golbabai, “Construction of a solitary wave solution for the generalized Zakharov equation by a variational iteration method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1003–1009, 2007.

O. P. Layeni, “A new rational auxiliary equation method and exact solutions of a generalized Zakharov system,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 2901–2907, 2009.

J. Zhang, “Variational approach to solitary wave solution of the generalized Zakharov equation,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1043–1046, 2007.

M. Javidi and A. Golbabai, “Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method,” Chaos, Solitons & Fractals, vol. 36, no. 2, pp. 309–313, 2008.

Y.-z. Li, K.-m. Li, and C. Lin, “Exp-function method for solving the generalized-Zakharov equations,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 197–201, 2008.

A. Borhanifar, M. M. Kabir, and L. M. Vahdat, “New periodic and soliton wave solutions for the generalized Zakharov system and -dimensional Nizhnik-Novikov-Veselov system,” Chaos, Solitons & Fractals, vol. 42, no. 3, pp. 1646–1654, 2009.

Y. Khan, N. Faraz, and A. Yildirim, “New soliton solutions of the generalized Zakharov equations using He's variational approach,” Applied Mathematics Letters, vol. 24, no. 6, pp. 965–968, 2011.

M. Song and Z. R. Liu, “Traveling wave solutions for the generalized Zakharov equations,” Mathematical Problems in Engineering, vol. 2012, Article ID 747295, 14 pages, 2012.

J. Li and Z. Liu, “Smooth and non-smooth traveling waves in a nonlinearly dispersive equation,” Applied Mathematical Modelling, vol. 25, no. 1, pp. 41–56, 2000.

J. Li and Y. Zhang, “Exact travelling wave solutions in a nonlinear elastic rod equation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 504–510, 2008.

M. Song, X. Hou, and J. Cao, “Solitary wave solutions and kink wave solutions for a generalized KDV-mKDV equation,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5942–5948, 2011.

M. Song, J. Cao, and X. Guan, “Application of the bifurcation method to the Whitham-Broer-Kaup-Like equations,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 688–696, 2012.