Traveling Wave Solutions for the Generalized Zakharov Equations

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.


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 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, 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.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 h 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.

Phase Portraits and Qualitative Analysis
We assume that the traveling wave solutions of 1.2 is of the form where ϕ ξ and ψ ξ are real functions; p, q, and k are real constants.Substituting 2.1 into 1.2 , we have

2.2
Integrating the second equation of 2.2 twice, and letting the first integral constant be zero, we have where g is 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 dϕ dξ y, dy dξ αϕ 3 − βϕ.

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

2.6
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 2.9 From the qualitative theory of dynamical systems, we know that: Therefore, we obtain the 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.
1 When h < 0 or h > h * , system 2.5 does not have any closed orbit.
1 When h −h * , 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 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.
1 When 0 < h < h * , 1.2 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, 1.2 has two periodic blow-up wave solutions (corresponding to the periodic orbits Γ 4 and Γ 5 in Figure 1).
3 When h h * , 1.2 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).
2 When h 0, 1.2 has two solitary wave solutions (corresponding to the homoclinic orbits Γ 10 and Γ 11 in Figure 1).
3 When h > 0, 1.2 has two periodic wave solutions (corresponding to the periodic orbit Γ 12 in Figure 1).

Traveling Wave Solutions and Their Relations
Firstly, we will obtain the explicit expressions of traveling wave solutions for the 1.2 when α > 0 and β > 0.

3.3
Noting that 2.1 and 2.3 , we get the following periodic wave solutions: where η px qt and ξ k x − 2pt .
2 From the phase portrait, we note that there are two special orbits Γ 4 and Γ 5 , which have the same hamiltonian as that of the center point 0, 0 .In ϕ, y plane the expressions of the orbits are given as where ϕ 5 − 2β/α and ϕ 6 2β/α.Substituting 3.5 into dϕ/dξ y, and integrating them along the two orbits Γ 4 and Γ 5 , it follows that Completing above integrals we obtain ϕ ± 2β α csc βξ.

3.7
Noting 2.1 and 2.3 , we get the following periodic blow-up wave solutions: where η px qt and ξ k x − 2pt .
3 From the phase portrait, we see that there are two heteroclinic orbits Γ 6 and Γ 7 connected at saddle points ϕ − , 0 and ϕ , 0 .In ϕ, y plane the expressions of the heteroclinic orbits are given as 3.9 Substituting 3.9 into dϕ/dξ y, and integrating them along the heteroclinic orbits Γ 6 and Γ 7 , it follows that

3.10
Completing above integrals we obtain

3.11
Noting 2.1 and 2.3 , we get the following kink profile solitary wave solutions:

3.12
Mathematical Problems in Engineering and unbounded wave solutions

3.13
where η px qt and ξ k x − 2pt .Secondly, we will obtain the explicit expressions of traveling wave solutions for 1.2 when α < 0 and β < 0.

3.15
Completing above integrals we obtain
Noting 2.1 and 2.3 , we get the following periodic wave solutions:
2 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 where ϕ 11 − 2β/α and ϕ 12 2β/α.Substituting 3.18 into dϕ/dξ y, and integrating them along the orbits Γ 10 and Γ 11 , we have  where η px qt and ξ k x − 2pt .
3 From the phase portrait, we see that there is a closed orbit Γ 12 passing the points ϕ 13 , 0 and ϕ 14 , 0 .In ϕ, y plane the expressions of the closed orbits are given as where  Finally, we will show that the periodic wave solutions u 2 x, t evolute into the kinkprofile solitary wave solutions u 4 x, t when the Hamiltonian h → h * − corresponding to the changes of phase orbits of Figure 1 as h varies .We take some suitable choices of the parameters, such as λ 1, k 1, p 1, q 1, g 2, 3.26 as an illustrative sample and draw their plots see Figures 2 and 3 .

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.

Figure 2 :
Figure 2: The real part of the periodic wave solution u 2 x, t evolutes into the kink-profile solitary wave solutions u 4 x, t with the conditions 3.26 .a h 0.5; b h 0.749; c h 0.75.

Figure 3 :
Figure 3: The imaginary part of the periodic wave solution u 2 x, t evolutes into the kink-profile solitary wave solutions u 4 x, t with the conditions 3.26 .a h 0.5; b h 0.749; c h 0.75.
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 .