Traveling Wave Solutions for the Generalized Zakharov Equations

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Volume 2012, Article ID 747295,14pages doi:10.1155/2012/747295

Research Article

Traveling Wave Solutions for the Generalized Zakharov Equations

Ming Song

^{1, 2}

and Zhengrong Liu

¹

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

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

Correspondence should be addressed to Ming Song,songming12 [email protected] Received 12 March 2012; Accepted 30 March 2012

Academic Editor: Anders Eriksson

Copyrightq2012 M. Song and Z. 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.

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

iutuxx−uv0, v_tt−v_xx

|u|²

xx0, 1.1

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 plasma2,3. 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 in1.1. This is equivalent to saying that1.1 is a simplified model of strong Langmuir turbulence. Thus we have to generalize1.1by

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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, 1.1are 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 as8

iu_tu_xx−2λ|u|²u2uv0, vtt−vxx

|u|²

xx0. 1.2

Malomed et al.8analyzed internal vibrations of a solitary wave in1.2by means of a variational approach. Wang and Li9obtained a number of periodic wave solutions of1.2 by using extended F-expansion method. Javidi and Golbabai10used the He’s variational iteration method to obtain solitary wave solutions of 1.2. Zhang 11 obtained the exact traveling wave solutions of1.2by using the direct algebraic method. Zhang12used He’s semi-inverse method to search for solitary wave solutions of1.2. Javidi and Golbabai13 obtained the exact and numerical solutions of1.2by using the variational iteration method.

Li et al.14used the Exp-function method to seek exact solutions of1.2. Borhanifar et al.

15obtained the generalized solitary solutions and periodic solutions of 1.2by using the Exp-function method. Khan et al.16used He’s variational approach to obtain new soliton solutions of1.2.

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 of1.2are given under diﬀerent parameter conditions. The relations between the traveling wave solutions and the Hamiltonianhare presented. InSection 3, we obtain a number of traveling wave solutions of1.2and give the relations of the traveling wave solutions. A short conclusion will be given inSection 4.

2. Phase Portraits and Qualitative Analysis

We assume that the traveling wave solutions of1.2is of the form ux, t e^iηϕξ, vx, t ψξ, ηpxqt, ξk

x−2pt

, 2.1

whereϕξandψξare real functions;p, q, andkare real constants.

Substituting2.1into1.2, we have k²ϕ2ϕψ−

p²q

ϕ−2λϕ³0, k²

4p²−1

ψk² ϕ²

0.

2.2

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Integrating the second equation of2.2twice, and letting the first integral constant be zero, we have

ψ ϕ²

1−4p² g, p /1

2, 2.3

wheregis integral constant.

Substituting2.3into the first equation of2.2, we have k²ϕ

2g−p²−q ϕ2

1 1−4p² −λ

ϕ³0. 2.4

Lettingϕy,α 2/k²λ−1/1−4p², andβ 2g−p²−q/k², then we get the following planar system

dϕ dξ y, dy

dξ αϕ³−βϕ.

2.5

Obviously, the above system2.5is a Hamiltonian system with Hamiltonian function H

ϕ, y

y²−1

2αϕ⁴βϕ². 2.6

In order to investigate the phase portrait of2.5, set f

ϕ

αϕ³−βϕ. 2.7 Obviously,fϕhas three zero points,ϕ₋,ϕ0, andϕ, which are given as follows:

ϕ₋− β

α, ϕ00, ϕ β

α. 2.8

Letting ϕi,0 be one of the singular points of system 2.5, then the characteristic values of the linearized system of system2.5at the singular pointsϕi,0are

λ_± ± f ϕ_i

. 2.9

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

1iffϕi>0,ϕi,0is a saddle point;

2iffϕi<0,ϕi,0is a center point;

3iffϕi 0,ϕi,0is a degenerate saddle point;

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Γ1

Γ2

Γ3

Γ4 Γ5

Γ6

ϕ1 ϕ2 ϕ3 ϕ4

ϕ5 ϕ6

ϕ− ϕ

y

Γ7

ϕ+

a

y

Γ8 Γ9

ϕ7 ϕ8 ϕ9

Γ10 Γ12 Γ11

ϕ10

ϕ11 ϕ12

ϕ13 ϕ14

ϕ

b

Figure 1: The phase portraits of system2.5.aα >0, β >0,bα <0, β <0.

Therefore, we obtain the phase portraits of system2.5inFigure 1.

Let

H ϕ, y

h, 2.10 wherehis Hamiltonian.

Next, we consider the relations between the orbits of2.5and the Hamiltonianh.

Set

h^∗H

ϕ,0H

ϕ₋,0 β²

2|α|. 2.11

According toFigure 1, we get the following propositions.

Proposition 2.1. Suppose thatα >0 andβ >0, we have the following.

1Whenh <0 orh > h^∗, system2.5does not have any closed orbit.

2When 0< h < h^∗, system2.5has three periodic orbitsΓ1,Γ2, andΓ3. 3Whenh0, system2.5has two periodic orbitsΓ4andΓ5.

4Whenhh^∗, system2.5has two heteroclinic orbitsΓ6andΓ7. Proposition 2.2. Suppose thatα <0 andβ <0, we have the following.

1Whenh−h^∗, system2.5does not have any closed orbit.

2When−h^∗< h <0, system2.5has two periodic orbitsΓ8andΓ9. 3Whenh0, system2.5has two homoclinic orbitsΓ10andΓ11. 4Whenh >0, system2.5has a periodic orbitΓ12.

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

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partial diﬀerential system. According to the above analysis, we have the following propositions.

Proposition 2.3. Ifα >0 andβ >0, we have the following.

1When 0 < h < h^∗,1.2has two periodic wave solutions (corresponding to the periodic orbitΓ2inFigure 1) and two periodic blow-up wave solutions (corresponding to the periodic orbitsΓ1andΓ3inFigure 1).

2Whenh0,1.2has two periodic blow-up wave solutions (corresponding to the periodic orbitsΓ4andΓ5inFigure 1).

3Whenhh^∗,1.2has two kink-profile solitary wave solutions and two unbounded wave solutions (corresponding to the heteroclinic orbitsΓ6andΓ7inFigure 1).

Proposition 2.4. Ifα <0 andβ <0, we have the following.

1When−h^∗ < h < 0,1.2has two periodic wave solutions (corresponding to the periodic orbitsΓ8andΓ9inFigure 1).

2Whenh0,1.2has two solitary wave solutions (corresponding to the homoclinic orbits Γ10andΓ11inFigure 1).

3Whenh >0,1.2has two periodic wave solutions (corresponding to the periodic orbitΓ12

inFigure 1).

3. Traveling Wave Solutions and Their Relations

Firstly, we will obtain the explicit expressions of traveling wave solutions for the1.2when α >0 andβ >0.

1 From the phase portrait, we note that there are three periodic orbitsΓ1,Γ2, and Γ3 passing the pointsϕ1,0,ϕ2,0,ϕ3,0, andϕ4,0. Inϕ, yplane the expressions of the orbits are given as

y± α

2 ϕ−ϕ₁

ϕ−ϕ₂

ϕ−ϕ₃ ϕ−ϕ₄

, 3.1

whereϕ1 −

β β²−2αh/α,ϕ2 −

β− β²−2αh/α,ϕ3

β− β²−2αh/α, ϕ₄

β β²−2αh/α, and 0< h < h^∗.

Substituting3.1into dϕ/dξyand integrating them alongΓ1, Γ2, andΓ3, we have

± _∞

ϕ

1 s−ϕ₁

s−ϕ₂

s−ϕ₃

s−ϕ₄ds α

2 _ξ

0

ds,

± _ϕ

0

1 s−ϕ₁

s−ϕ₂

s−ϕ₃

s−ϕ₄ds α

2 _ξ

0

ds.

3.2

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Completing above integrals we obtain

ϕ± ϕ4

sn ϕ₄

α/2ξ, ϕ₃/ϕ₄, ϕ±ϕ3sn

ϕ₄

α 2ξ,ϕ₃

ϕ₄

.

3.3

Noting that2.1and2.3, we get the following periodic wave solutions:

u₁x, t ± e^iηϕ₄ sn

ϕ₄

α/2ξ, ϕ₃/ϕ₄, v₁x, t ϕ²₄

1−4p² sn

ϕ₄

α/2ξ, ϕ₃/ϕ₄₂ g, u₂x, t ±e^iηϕ₃sn

ϕ₄

α 2ξ,ϕ3

ϕ4

,

v₂x, t

ϕ₃sn ϕ₄

α/2ξ, ϕ₃/ϕ₄₂

1−4p² g,

3.4

whereηpxqtandξkx−2pt.

2From the phase portrait, we note that there are two special orbitsΓ4andΓ5, which have the same hamiltonian as that of the center point0,0. Inϕ, yplane the expressions of the orbits are given as

y± α

2ϕ ϕ−ϕ₅ ϕ−ϕ₆

, 3.5

whereϕ₅−

2β/αandϕ₆ 2β/α.

Substituting3.5into dϕ/dξ y, and integrating them along the two orbitsΓ4and Γ5, it follows that

± _∞

ϕ

1 s s−ϕ₅

s−ϕ₆ds α

2 _ξ

0

ds. 3.6

ϕ± 2β

αcsc βξ. 3.7

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Noting2.1and2.3, we get the following periodic blow-up wave solutions:

u₃x, t ±e^iη 2β

αcsc βξ,

v3x, t 2β csc

βξ₂ α

1−4p² g,

3.8

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

y± α

2 ϕ−ϕ₋₂

ϕ−ϕ₂

. 3.9

Substituting3.9into dϕ/dξ y, and integrating them along the heteroclinic orbits Γ6andΓ7, it follows that

± _ϕ

0

1

s−ϕ₋

ϕ−sds α

2 _ξ

0

ds,

± _∞

ϕ

1

s−ϕ₋

s−ϕds α

2 _ξ

0

ds.

3.10

ϕ± β

αtanh β

2ξ, ϕ±

β αcoth

β 2ξ.

3.11

Noting2.1and2.3, we get the following kink profile solitary wave solutions:

u₄x, t ±e^iη β

αtanh β

2ξ, v₄x, t β

tanh βξ2

α

1−4p² g,

3.12

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and unbounded wave solutions

u₅x, t ±e^iη β

αcoth β

2ξ,

v₅x, t β coth

βξ₂ α

1−4p² g,

3.13

Secondly, we will obtain the explicit expressions of traveling wave solutions for1.2 whenα <0 andβ <0.

1From the phase portrait, we see that there are two closed orbitsΓ8andΓ9passing the points ϕ7,0, ϕ8,0, ϕ9,0, andϕ10,0. Inϕ, y plane the expressions of the closed orbits are given as

y±

−α

2 ϕ−ϕ₇

ϕ−ϕ₈

ϕ−ϕ₉

ϕ₁₀−ϕ

, 3.14

whereϕ₇ −

β− β²−2αh/α,ϕ₈ −

β β²−2αh/α,ϕ₉

β β²−2αh/α, ϕ10

β− β²−2αh/α, and−h^∗< h <0.

Substituting3.14into dϕ/dξy, and integrating them alongΓ8andΓ9, we have

± _ϕ

ϕ7

1 ϕ10−s

ϕ9−s

ϕ8−s s−ϕ7

ds

−α 2

_ξ

0

ds,

± _ϕ

ϕ10

1 s−ϕ₇

s−ϕ₈

s−ϕ₉

ϕ₁₀−sds

−α 2

_ξ

0

ds.

3.15

ϕ

ϕ₁₀−ϕ₈ ϕ₇

ϕ₈−ϕ₇ ϕ₁₀

sn ω

−α/2ξ, κ2

ϕ₁₀−ϕ₈

ϕ₈−ϕ₇ sn

ω

−α/2ξ, κ₂ ,

ϕ ϕ²₁₀−

ϕ²₁₀−ϕ²₉

⎛

⎜⎝sn

⎛

⎜⎝ϕ10

−α

2ξ, ϕ²₁₀−ϕ²₉ ϕ₁₀

⎞

⎟⎠

⎞

⎟⎠

2

,

3.16

whereω

ϕ10−ϕ₈ϕ9−ϕ₇/2 andκ

ϕ10−ϕ₉ϕ8−ϕ₇/ϕ10−ϕ₈ϕ9−ϕ₇.

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Noting2.1and2.3, we get the following periodic wave solutions:

u₆x, t e^iη

ϕ₁₀−ϕ₈ ϕ₇

ϕ₈−ϕ₇ ϕ₁₀

sn ω

−α/2ξ, κ₂ ϕ₁₀−ϕ₈

ϕ₈−ϕ₇ sn

ω

−α/2ξ, κ₂ ,

v₆x, t

ϕ10−ϕ₈ϕ7 ϕ8−ϕ₇ϕ10

sn

ω

−α/2ξ, κ₂₂ 1−4p²

ϕ₁₀−ϕ₈ ϕ8−ϕ₇ sn

ω

−α/2ξ, κ₂₂ g,

u₇x, t e^iη ϕ²₁₀−

ϕ²₁₀−ϕ²₉

⎛

⎜⎝sn

⎛

⎜⎝ϕ₁₀ −α/2ξ, ϕ²₁₀−ϕ²₉ ϕ10

⎞

⎟⎠

⎞

⎟⎠

2

,

v7x, t ϕ²₁₀−

ϕ²₁₀−ϕ²₉ sn

ϕ10

−α/2ξ, ϕ²₁₀−ϕ²₉/ϕ10

2

1−4p² g,

3.17

2From the phase portrait, we see that there are two symmetric homoclinic orbitsΓ10

andΓ11connected at the saddle point0,0. Inϕ, yplane the expressions of the homoclinic orbits are given as

y±

−α

2ϕ ϕ−ϕ₁₁

ϕ₁₂−ϕ

, 3.18

whereϕ₁₁−

2β/αandϕ₁₂ 2β/α.

Substituting3.18into dϕ/dξy, and integrating them along the orbitsΓ10andΓ11, we have

± _ϕ

ϕ11

1 s s−ϕ11

ϕ12−sds

−α 2

_ξ

0

ds,

± _ϕ

ϕ12

1 s s−ϕ₁₁

ϕ₁₂−sds

−α 2

_ξ

0

ds.

3.19

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ϕ 2β

αsech −βξ, ϕ−

2β

αsech −βξ.

3.20

Noting2.1and2.3, we get the following solitary wave solutions:

u8x, t e^iη 2β

αsech −βξ,

v₈x, t 2β sech

−βξ₂ α

1−4p² g, u9x, t −e^iη

2β

αsech −βξ,

v₉x, t 2β sech

−βξ₂ α

1−4p² g,

3.21

3From the phase portrait, we see that there is a closed orbitΓ12 passing the points ϕ13,0andϕ14,0. Inϕ, yplane the expressions of the closed orbits are given as

y±

−α

2 ϕ₁₄−ϕ

ϕ−ϕ₁₃

ϕ−c₁ ϕ−c₁

, 3.22

whereϕ14

β− β²−2αh/α,ϕ13 −

β− β²−2αh/α,c1 i

−β− β²−2αh/α, c₁−i

−β− β²−2αh/α, andh >0.

Substituting3.22into dϕ/dξy, and integrating them along the orbitΓ12, we have

± _ϕ

ϕ13

1 ϕ₁₄−s

s−ϕ₁₃

s−c₁s−c₁ds

−α 2

_ξ

0

ds,

± _ϕ₁₄

ϕ

1 ϕ₁₄−s

s−ϕ₁₃

s−c₁s−c₁ds

−α 2

_ξ

0

ds.

3.23

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ϕϕ13cn −βξ,−ϕ13

α 2β

,

ϕϕ₁₄cn −βξ, ϕ14

α 2β

.

3.24

Noting2.1and2.3, we get the following periodic wave solutions:

u10x, t e^iηϕ13cn −βξ,−ϕ13

α 2β

,

v₁₀x, t

ϕ₁₃cn

−βξ,−ϕ13

α/2β2

1−4p² g,

u11x, t e^iηϕ14cn −βξ, ϕ14 α/2β

,

v₁₁x, t

ϕ14cn

−βξ, ϕ14

α/2β₂

1−4p² g,

3.25

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

1Letting h → h^∗−, it follows that ϕ₄ →

β/α,ϕ₃ →

β/α,ϕ₃/ϕ₄ → 1 and

sn

βξ,1 tanh

βξ. Therefore, we obtainu₁x, t → u₅x, t,v₁x, t → v₅x, t, u2x, t → u4x, tandv2x, t → v4x, t.

2Lettingh → 0, it follows thatϕ₄ →

2β/α,ϕ₃ → 0,ϕ₃/ϕ₄ → 0 and sn βξ,0 sin

βξ. Therefore, we obtainu1x, t → u3x, tandv1x, t → v3x, t.

3Lettingh → 0−, it follows thatϕ10 →

2β/α,ϕ9 → 0,ϕ8 → 0,ϕ7 → − 2β/α,

ω →

β/2α,k → 1 and sn

−β/2ξ,1 tanh

−β/2ξ. Therefore, we obtain u₆x, t → u₈x, tandv₆x, t → v₈x, t.

4Lettingh → 0−, it follows thatϕ₁₀ →

2β/α,ϕ₉ → 0,ϕ₈ → 0,ϕ₇ → − 2β/α, ϕ²₁₀−ϕ²₉/ϕ₁₀ → 1 and sn

−βξ,1 tanh

−βξ. Therefore, we obtainu₇x, t → u9x, tandv7x, t → v9x, t.

5Lettingh → 0, it follows thatϕ₁₄ →

2β/α,ϕ₁₃ → −

2β/α,−ϕ13

α/2β → 1, ϕ14

α/2β → 1 and cn

−βξ,1 sech

−βξ. Therefore, we obtain u10x, t → u₉x, t,v₁₀x, t → v₉x, t,u₁₁x, t → u₈x, tandv₁₁x, t → v₈x, t.

Finally, we will show that the periodic wave solutionsu2x, tevolute into the kink- profile solitary wave solutions u₄x, twhen the Hamiltonianh → h^∗−corresponding to

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10

15 x

0

5 t

0.5

10

5 0

−0.5

a

10

15 0 5

0.5 10

5 0

−0.5

x

t

b

x

t 10

15 0 5

0.5 10

5 0

−0.5

c

Figure 2: The real part of the periodic wave solutionu2x, tevolutes into the kink-profile solitary wave solutionsu4x, twith the conditions3.26.ah0.5;bh0.749;ch0.75.

the changes of phase orbits of Figure 1as hvaries. We take some suitable choices of the parameters, such as

λ1, k1, p1, q1, g2, 3.26

as an illustrative sample and draw their plotssee Figures2and3.

4. Conclusion

In this paper, we obtain phase portraits for the corresponding traveling wave system of1.2 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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10

15 0 5

0.5

10

5 0

−0.5

x

t

a

10

15 0 5

0.5 10

5 0

−0.5

x

t

b

10

15 0 5

0.5 10

5 0

−0.5

x

t

c

Figure 3: The imaginary part of the periodic wave solutionu2x, tevolutes into the kink-profile solitary wave solutionsu4x, twith the conditions3.26.ah0.5;bh0.749;ch0.75.

Acknowledgment

Research is supported by the National Natural Science Foundation of China Grant no.

11171115and the Natural Science Foundation of Yunnan ProvinceGrant no. 2010ZC154.

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22 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.

23 C. E. Ye and W. G. Zhang, “New explicit solutions for21-dimensional soliton equation,” Chaos, Solitons & Fractals, vol. 44, pp. 1063–1069, 2011.

24 M. Song, J. Cao, and X. L. Guan, “Application of the bifurcation method to the Whitham-Broer-Kaup- Like equations,” Mathematical and Computer Modelling, vol. 55, pp. 688–696, 2012.

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