# Traveling Wave Solutions for the Generalized Zakharov Equations

Authors: Ming Song; Zhengrong Liu
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 1. Introduction

The Zakharov equations(1.1)iut+uxx-uv=0,vtt-vxx+(|u|2)xx=0,
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]
(1.2)iut+uxx-2λ|u|2u+2uv=0,vtt-vxx+(|u|2)xx=0.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 Section2, 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.


## 2. Phase Portraits and Qualitative Analysis

We assume that the traveling wave solutions of (1.2) is of the form
(2.1)u(x,t)=eiηφ(ξ),v(x,t)=ψ(ξ),η=px+qt,ξ=k(x-2pt),
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)k2φ′′+2φψ-(p2+q)φ-2λφ3=0,k2(4p2-1)ψ′′+k2(φ2)′′=0.Integrating the second equation of (2.2) twice, and letting the first integral constant be zero, we have
(2.3)ψ=φ21-4p2+g,p≠12,
where g is integral constant.Substituting (2.3) into the first equation of (2.2), we have
(2.4)k2φ′′+(2g-p2-q)φ+2(11-4p2-λ)φ3=0.Lettingφ′=y, α=(2/k2)(λ-1/(1-4p2)), and β=(2g-p2-q)/k2, then we get the following planar system
(2.5)dφdξ=y,dydξ=αφ3-βφ.Obviously, the above system (2.5) is a Hamiltonian system with Hamiltonian function
(2.6)H(φ,y)=y2-12αφ4+βφ2.In order to investigate the phase portrait of (2.5), set
(2.7)f(φ)=αφ3-βφ.Obviously,f(φ) has three zero points, φ-, φ0, and φ+, which are given as follows:
(2.8)φ-=-βα,φ0=0,φ+=βα.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
(2.9)λ±=±f′(φi).From the qualitative theory of dynamical systems, we know that:(1)
iff′(φi)>0, (φi,0) is a saddle point;(2)
iff′(φi)<0, (φi,0) is a center point;(3)
iff′(φi)=0, (φi,0) is a degenerate saddle point;Therefore, we obtain the phase portraits of system (2.5) in Figure 1.The phase portraits of system (2.5). (a) α>0,β>0, (b) α<0,β<0.
(a)
(b)Let(2.10)H(φ,y)=h,
where h is Hamiltonian.Next, we consider the relations between the orbits of (2.5) and the Hamiltonian h.Set(2.11)h*=|H(φ+,0)|=|H(φ-,0)|=β22|α|.
According to Figure 1, we get the following propositions.Proposition 2.1.
Suppose thatα>0 and β>0, we have the following. (1)
Whenh<0 or h>h*, system (2.5) does not have any closed orbit.(2)
When0<h<h*, system (2.5) has three periodic orbits Γ1, Γ2, and Γ3.(3)
Whenh=0, system (2.5) has two periodic orbits Γ4 and Γ5.(4)
Whenh=h*, system (2.5) has two heteroclinic orbits Γ6 and Γ7.Proposition 2.2.
Suppose thatα<0 and β<0, we have the following.(1)
Whenh⩽-h*, system (2.5) does not have any closed orbit.(2)
When-h*<h<0, system (2.5) has two periodic orbits Γ8 and Γ9.(3)
Whenh=0, system (2.5) has two homoclinic orbits Γ10 and Γ11.(4)
Whenh>0, system (2.5) has a periodic orbit Γ12.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.
Ifα>0 and β>0, we have the following.(1)
When0<h<h*, (1.2) has two periodic wave solutions (corresponding to the periodic orbit Γ2 in Figure 1) and two periodic blow-up wave solutions (corresponding to the periodic orbits Γ1 and Γ3 in Figure 1).(2)
Whenh=0, (1.2) has two periodic blow-up wave solutions (corresponding to the periodic orbits Γ4 and Γ5 in Figure 1).(3)
Whenh=h*, (1.2) has two kink-profile solitary wave solutions and two unbounded wave solutions (corresponding to the heteroclinic orbits Γ6 and Γ7 in Figure 1).Proposition 2.4.
Ifα<0 and β<0, we have the following.(1)
When-h*<h<0, (1.2) has two periodic wave solutions (corresponding to the periodic orbits Γ8 and Γ9 in Figure 1).(2)
Whenh=0, (1.2) has two solitary wave solutions (corresponding to the homoclinic orbits Γ10 and Γ11 in Figure 1).(3)
Whenh>0, (1.2) has two periodic wave solutions (corresponding to the periodic orbit Γ12 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 α>0 and β>0.(
1
) From the phase portrait, we note that there are three periodic orbits Γ1, Γ2, and Γ3 passing the points (φ1,0),(φ2,0),(φ3,0), and (φ4,0). In (φ,y) plane the expressions of the orbits are given as
(3.1)y=±α2(φ-φ1)(φ-φ2)(φ-φ3)(φ-φ4),
where φ1=-(β+β2-2αh)/α, φ2=-(β-β2-2αh)/α, φ3=(β-β2-2αh)/α, φ4=(β+β2-2αh)/α, and 0<h<h*.Substituting (3.1) into dφ/dξ=y and integrating them along Γ1,Γ2, and Γ3, we have
(3.2)±∫φ∞‍1(s-φ1)(s-φ2)(s-φ3)(s-φ4)ds=α2∫0ξ‍ds,±∫0φ‍1(s-φ1)(s-φ2)(s-φ3)(s-φ4)ds=α2∫0ξ‍ds.Completing above integrals we obtain(3.3)φ=±φ4sn(φ4α/2ξ,φ3/φ4),φ=±φ3sn(φ4α2ξ,φ3φ4).Noting that (2.1) and (2.3), we get the following periodic wave solutions:
(3.4)u1(x,t)=±eiηφ4sn(φ4α/2ξ,φ3/φ4),v1(x,t)=φ42(1-4p2)(sn(φ4α/2ξ,φ3/φ4))2+g,u2(x,t)=±eiηφ3sn(φ4α2ξ,φ3φ4),v2(x,t)=(φ3sn(φ4α/2ξ,φ3/φ4))21-4p2+g,
where η=px+qt and ξ=k(x-2pt).(
2
) From the phase portrait, we note that there are two special orbits Γ4 and Γ5, 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
(3.5)y=±α2φ(φ-φ5)(φ-φ6),
where φ5=-2β/α and φ6=2β/α.Substituting (3.5) into dφ/dξ=y, and integrating them along the two orbits Γ4 and Γ5, it follows that
(3.6)±∫φ+∞‍1s(s-φ5)(s-φ6)ds=α2∫0ξ‍ds.Completing above integrals we obtain(3.7)φ=±2βαcscβξ.Noting (2.1) and (2.3), we get the following periodic blow-up wave solutions:
(3.8)u3(x,t)=±eiη2βαcscβξ,v3(x,t)=2β(cscβξ)2α(1-4p2)+g,
where η=px+qt and ξ=k(x-2pt).(
3
) From the phase portrait, we see that there are two heteroclinic orbits Γ6 and Γ7 connected at saddle points (φ-,0) and (φ+,0). In (φ,y) plane the expressions of the heteroclinic orbits are given as
(3.9)y=±α2(φ-φ-)2(φ-φ+)2.Substituting (3.9) into dφ/dξ=y, and integrating them along the heteroclinic orbits Γ6 and Γ7, it follows that
(3.10)±∫0φ‍1(s-φ-)(φ+-s)ds=α2∫0ξ‍ds,±∫φ+∞‍1(s-φ-)(s-φ+)ds=α2∫0ξ‍ds.Completing above integrals we obtain(3.11)φ=±βαtanhβ2ξ,φ=±βαcoth⁡β2ξ.Noting (2.1) and (2.3), we get the following kink profile solitary wave solutions:
(3.12)u4(x,t)=±eiηβαtanhβ2ξ,v4(x,t)=β(tanhβξ)2α(1-4p2)+g,
and unbounded wave solutions
(3.13)u5(x,t)=±eiηβαcoth⁡β2ξ,v5(x,t)=β(coth⁡βξ)2α(1-4p2)+g,
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.(
1
) From the phase portrait, we see that there are two closed orbits Γ8 and Γ9 passing the points (φ7,0),(φ8,0),(φ9,0), and (φ10,0). In (φ,y) plane the expressions of the closed orbits are given as
(3.14)y=±-α2(φ-φ7)(φ-φ8)(φ-φ9)(φ10-φ),
where φ7=-(β-β2-2αh)/α, φ8=-(β+β2-2αh)/α, φ9=(β+β2-2αh)/α, φ10=(β-β2-2αh)/α, and -h*<h<0.Substituting (3.14) into dφ/dξ=y, and integrating them along Γ8 and Γ9, we have
(3.15)±∫φ7φ‍1(φ10-s)(φ9-s)(φ8-s)(s-φ7)ds=-α2∫0ξ‍ds,±∫φ10φ‍1(s-φ7)(s-φ8)(s-φ9)(φ10-s)ds=-α2∫0ξ‍ds.Completing above integrals we obtain(3.16)φ=(φ10-φ8)φ7+(φ8-φ7)φ10(sn(ω-(α/2)ξ,κ))2φ10-φ8+(φ8-φ7)(sn(ω-(α/2)ξ,κ))2,φ=φ102-(φ102-φ92)(sn(φ10-α2ξ,φ102-φ92φ10))2,
where ω=(φ10-φ8)(φ9-φ7)/2 and κ=(φ10-φ9)(φ8-φ7)/(φ10-φ8)(φ9-φ7).Noting (2.1) and (2.3), we get the following periodic wave solutions:
(3.17)u6(x,t)=eiη((φ10-φ8)φ7+(φ8-φ7)φ10(sn(ω-(α/2)ξ,κ))2)φ10-φ8+(φ8-φ7)(sn(ω-(α/2)ξ,κ))2,v6(x,t)=((φ10-φ8)φ7+(φ8-φ7)φ10(sn(ω-(α/2)ξ,κ))2)2(1-4p2)(φ10-φ8+(φ8-φ7)(sn(ω-(α/2)ξ,κ))2)2+g,u7(x,t)=eiηφ102-(φ102-φ92)(sn(φ10-(α/2)ξ,φ102-φ92φ10))2,v7(x,t)=φ102-(φ102-φ92)(sn(φ10-(α/2)ξ,φ102-φ92/φ10))21-4p2+g,
where η=px+qt and ξ=k(x-2pt).(
2
) From the phase portrait, we see that there are two symmetric homoclinic orbits Γ10 and Γ11 connected at the saddle point (0,0). In (φ,y) plane the expressions of the homoclinic orbits are given as
(3.18)y=±-α2φ(φ-φ11)(φ12-φ),
where φ11=-2β/α and φ12=2β/α.Substituting (3.18) into dφ/dξ=y, and integrating them along the orbits Γ10 and Γ11, we have
(3.19)±∫φ11φ‍1s(s-φ11)(φ12-s)ds=-α2∫0ξ‍ds,±∫φ12φ‍1s(s-φ11)(φ12-s)ds=-α2∫0ξ‍ds.Completing above integrals we obtain(3.20)φ=2βαsech-βξ,φ=-2βαsech-βξ.
Noting (2.1) and (2.3), we get the following solitary wave solutions:
(3.21)u8(x,t)=eiη2βαsech-βξ,v8(x,t)=2β(sech-βξ)2α(1-4p2)+g,u9(x,t)=-eiη2βαsech-βξ,v9(x,t)=2β(sech-βξ)2α(1-4p2)+g,
where η=px+qt and ξ=k(x-2pt).(
3
) From the phase portrait, we see that there is a closed orbit Γ12 passing the points (φ13,0) and (φ14,0). In (φ,y) plane the expressions of the closed orbits are given as
(3.22)y=±-α2(φ14-φ)(φ-φ13)(φ-c1)(φ-c¯1),
where φ14=(β-β2-2αh)/α, φ13=-(β-β2-2αh)/α, c1=i(-β-β2-2αh)/α, c¯1=-i(-β-β2-2αh)/α, and h>0.Substituting (3.22) into dφ/dξ=y, and integrating them along the orbit Γ12, we have
(3.23)±∫φ13φ‍1(φ14-s)(s-φ13)(s-c1)(s-c¯1)ds=-α2∫0ξ‍ds,±∫φφ14‍1(φ14-s)(s-φ13)(s-c1)(s-c¯1)ds=-α2∫0ξ‍ds.Completing above integrals we obtain(3.24)φ=φ13cn(-βξ,-φ13α2β),φ=φ14cn(-βξ,φ14α2β).Noting (2.1) and (2.3), we get the following periodic wave solutions:
(3.25)u10(x,t)=eiηφ13cn(-βξ,-φ13α2β),v10(x,t)=(φ13cn(-βξ,-φ13α/2β))21-4p2+g,u11(x,t)=eiηφ14cn(-βξ,φ14α/2β),v11(x,t)=(φ14cn(-βξ,φ14α/2β))21-4p2+g,
where η=px+qt and ξ=k(x-2pt).Thirdly, we will give the relations of the traveling wave solutions.(1)
Lettingh→h*-, it follows that φ4→β/α, φ3→β/α, φ3/φ4→1 and sn(βξ,1)=tanhβξ. Therefore, we obtain u1(x,t)→u5(x,t), v1(x,t)→v5(x,t), u2(x,t)→u4(x,t) and v2(x,t)→v4(x,t).(2)
Lettingh→0+, it follows that φ4→2β/α, φ3→0, φ3/φ4→0 and sn(βξ,0)=sinβξ. Therefore, we obtain u1(x,t)→u3(x,t) and v1(x,t)→v3(x,t).(3)
Lettingh→0-, it follows that φ10→2β/α, φ9→0, φ8→0, φ7→-2β/α, ω→β/2α, k→1 and sn((-β/2)ξ,1)=tanh(-β/2)ξ. Therefore, we obtain u6(x,t)→u8(x,t) and v6(x,t)→v8(x,t).(4)
Lettingh→0-, it follows that φ10→2β/α, φ9→0, φ8→0, φ7→-2β/α, φ102-φ92/φ10→1 and sn(-βξ,1)=tanh-βξ. Therefore, we obtain u7(x,t)→u9(x,t) and v7(x,t)→v9(x,t).(5)
Lettingh→0+, it follows that φ14→2β/α, φ13→-2β/α, -φ13α/2β→1, φ14α/2β→1 and cn(-βξ,1)=sech-βξ. Therefore, we obtain u10(x,t)→u9(x,t), v10(x,t)→v9(x,t), u11(x,t)→u8(x,t) and v11(x,t)→v8(x,t).Finally, we will show that the periodic wave solutionsu2(x,t) evolute into the kink-profile solitary wave solutions u4(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
(3.26)λ=1,k=1,p=1,q=1,g=2,
as an illustrative sample and draw their plots (see Figures 2 and 3).The real part of the periodic wave solutionu2(x,t) evolutes into the kink-profile solitary wave solutions u4(x,t) with the conditions (3.26). (a) h=0.5; (b) h=0.749; (c) h=0.75.
(a)
(b)
(c)The imaginary part of the periodic wave solutionu2(x,t) evolutes into the kink-profile solitary wave solutions u4(x,t) with the conditions (3.26). (a) h=0.5; (b) h=0.749; (c) h=0.75.
(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.



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