Application of Bifurcation Method to the Generalized Zakharov Equations

and Applied Analysis 3 Substituting 2.1 into 1.3 , we have k2φ′′ 2φψ − ( p2 q ) φ − 2λφ2n 1 0, k2 ( 4p2 − 1 ) ψ ′′ k2 ( φ2n )′′ 0. 2.2 Integrating the second equation of 2.2 twice and letting the first integral constant be zero, we have ψ φ2n 1 − 4p2 g, p / 1 2 , 2.3 where g is the second integral constant. Substituting 2.3 into the first equation of 2.2 , we have k2φ′′ ( 2g − p2 − q ) φ 2 ( 1 1 − 4p2 − λ ) φ2n 1 0. 2.4 Letting φ′ y, α 2/k2 λ − 1/ 1 − 4p2 , and β 2g − p2 − q /k2, then we get the following planar system: dφ dξ y, dy dξ αφ2n 1 − βφ. 2.5 Obviously, system 2.5 is a Hamiltonian system with Hamiltonian function H ( φ, y ) y2 − α n 1 φ2n 2 βφ2. 2.6 In order to investigate the phase portrait of 2.5 , set f ( φ ) αφ2n 1 − βφ. 2.7 Obviously, f φ has three zero points, φ−, φ0, and φ , which are given as follows: φ− − ( β α )1/2n , φ0 0, φ ( β α )1/2n . 2.8 Letting φi, 0 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 φi, 0 are λ± ± √ f ′ ( φi ) . 2.9 4 Abstract and Applied Analysis


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 lowfrequency 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 ionacoustic 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 timeaveraged 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 In this work, we aim to study new generalized Zakharov equations.
where n 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 h 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.

Phase Portraits and Qualitative Analysis
We assume that the traveling wave solutions of 1.3 is of the form where ϕ ξ and ψ ξ are real functions, p, q, and k are real constants.
Integrating the second equation of 2.2 twice and letting the first integral constant be zero, we have where g is the second integral constant. Substituting 2.3 into the first equation of 2.2 , we have Letting ϕ y, α 2/k 2 λ − 1/ 1 − 4p 2 , and β 2g − p 2 − q /k 2 , then we get the following planar system: 2.5 Obviously, system 2.5 is a Hamiltonian system with Hamiltonian function In order to investigate the phase portrait of 2.5 , set Obviously, f ϕ has three zero points, ϕ − , ϕ 0 , and ϕ , which are given as follows: Letting ϕ i , 0 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 ϕ i , 0 are λ ± ± f ϕ i .  Figure 1: The bifurcation phase portraits of system 2.5 . a α > 0, β > 0, b α < 0, β 0, c α < 0, β < 0, d α > 0, and β 0.
From the qualitative theory of dynamical systems, we know that Therefore, we obtain the bifurcation phase portraits of system 2.5 in Figure 1. Let where h is Hamiltonian. Next, we consider the relations between the orbits of 2.5 and the Hamiltonian h. Set According to Figure 1, we get the following propositions. 2 when −h * < h < 0, system 2.5 has two periodic orbits Γ 8 and Γ 9 ; 3 when h 0, system 2.5 has two homoclinic orbits Γ 10 and Γ 11 ; 4 when h > 0, system 2.5 has a periodic orbit Γ 12 . Proposition 2.3. 1 When α < 0, β ≥ 0, and h > 0, system 2.5 has a periodic orbits.
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 α > 0 and β > 0, one has the following: 1 when 0 < h < h * , 1.3 has two periodic wave solutions (corresponding to the periodic orbit Γ 2 in Figure 1) and two periodic blow-up wave solutions (corresponding to the periodic orbits Γ 1 and Γ 3 in Figure 1); 2 when h 0, 1.3 has periodic blow-up wave solutions (corresponding to the periodic orbits Γ 4 and Γ 5 in Figure 1); 3 when h h * , 1.3 has two kink profile solitary wave solutions and two unbounded wave solutions (corresponding to the heteroclinic orbits Γ 6 and Γ 7 in Figure 1).

Exact Traveling Wave Solutions of 1.3
Firstly, we will obtain the explicit expressions of traveling wave solutions for 1.3 when α > 0 and β > 0. From the phase portrait, we note that there are two special orbits Γ 4 and Γ 5 , which have the same Hamiltonian with that of the center point 0, 0 . In ϕ, y plane, the expressions of the orbits are given as Abstract and Applied Analysis Substituting 3.1 into dϕ/dξ y and integrating them along the two orbits Γ 4 and Γ 5 , it follows that where η px qt and ξ k x − 2pt . Secondly, we will obtain the explicit expressions of traveling wave solutions for 1.3 when α < 0 and β < 0. From the phase portrait, we see that there are two symmetric homoclinic orbits Γ 10 and Γ 11 connected at the saddle point 0, 0 . In ϕ, y plane, the expressions of the homoclinic orbits are given as Abstract and Applied Analysis 7 Substituting 3.5 into dϕ/dξ y and integrating them along the orbits Γ 10 and Γ 11 , we have

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.