Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 972416, 7 pageshttp://dx.doi.org/10.1155/2013/972416
Research ArticleTopological Soliton Solution and Bifurcation Analysis ofthe Klein-Gordon-Zakharov Equation in (1 + 1)-Dimensions withPower Law Nonlinearity
Ming Song,1 Bouthina S. Ahmed,2 and Anjan Biswas3,4
1 Department of Mathematics, Yuxi Normal University, Yuxi, Yunnan 653100, China2Department of Mathematics, Girlsβ College, Ain Shams University, Cairo 11757, Egypt3Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
Correspondence should be addressed to Anjan Biswas; [email protected]
Received 22 November 2012; Accepted 11 December 2012
Academic Editor: Abdul Hamid Kara
Copyright Β© 2013 Ming Song et al. 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.
This paper addresses the Klein-Gordon-Zakharov equation with power law nonlinearity in (1 + 1)-dimensions. The integrabilityaspect as well as the bifurcation analysis is studied in this paper.The numerical simulations are also given where the finite differenceapproach was utilized. There are a few constraint conditions that naturally evolve during the course of derivation of the solitonsolutions. These constraint conditions must remain valid in order for the soliton solution to exist. For the bifurcation analysis, thephase portraits are also given.
1. Introduction
The theory of nonlinear evolution equations (NLEEs) hascome a long way in the past few decades [1β20]. Many of theNLEEs are pretty well known in the area of theoretical physicsand applied mathematics. A few of them are the nonlinearSchroΜdingerβs equation, Korteweg-de Vries (KdV) equation,sine-Gordon equationwhich appear in nonlinear optics, fluiddynamics, and theoretical physics, respectively. It is also verycommon to come across several combo NLEEs such as theSchroΜdinger-KdV equation, Klein-Gordon-Zakharov (KGZ)equation, andmany others that are also studied in the contextof applied mathematics and theoretical physics. This paper isgoing to focus on the KGZ equation that will be studied withpower law nonlinearity in (1 + 1)-dimensions.
The integrability aspects and the bifurcation analysis willbe the main focus of this paper. The ansatz method willbe applied to obtain the topological 1-soliton solution, alsoknown as the shock wave solution, to this equation. Theconstraint conditions will be naturally formulated in orderfor the soliton solution to exist. Subsequently, the bifurcation
analysis will be carried out for this paper. In this context,the phase portraits will be given. Additionally, other travelingwave solutions will be enumerated. Finally, the numericalsimulation to the equation will be given. The finite differencescheme will also be given.
2. Mathematical Analysis
The KGZ equation with power law nonlinearity in (1 + 1)-dimensions that are going to be studied in this paper is givenby [6]
ππ‘π‘β π2ππ₯π₯+ ππ + πππ + π
π
2π
π = 0, (1)
ππ‘π‘β π2ππ₯π₯= π(
π
2π
)π₯π₯, (2)
where π, π, π, π, and π are real valued constants. Additionallyπ(π₯, π‘) is a complex valued dependent variable and π(π₯, π‘) isa real valued dependent variable. This section will focus onextracting the shock wave solutions to the KGZ equation (1)and (2) that are also known as topological soliton solution.
2 Journal of Applied Mathematics
Therefore the starting hypothesis will be
π (π₯, π‘) = π΄1tanhπ1ππππ, (3)
π (π₯, π‘) = π΄2tanhπ2π, (4)
where
π = π΅ (π₯ β π£π‘) . (5)
Here, in (3) and (4)π΄1, π΄2and π΅ are free parameters, while π£
is the velocity of the soliton. The unknown exponents π1and
π2will be determined, in terms of π by the aid of balancing
principle. The phase component of (3) is given by
π = βπ π₯ + ππ‘ + π, (6)
where π represents the soliton frequency, π is the solitonwave number, and π is the phase constant. Substituting thehypothesis (3) and (4) into (1) and (2) yields
π1(π1β 1) (π£
2β π2) π΅2tanhπ1β2π
β 2ππ1(π£π β π
2π2) π΅ tanhπ1β1π
β {2π2
1(π£2β π2) π΅2+ π2+ πΎ2π 2} tanhπ1π
+ 2ππ1(π£π β π
2π2) π΅ tanhπ1+1π
+ π1(π1+ 1) (π£
2β π2) π΅2 tanhπ1+2π + π tanhπ1π
+ ππ΄2tanhπ1+π2π + ππ΄2π
1tanh(2π+1)π1π = 0,
(7)
π2(π2β 1) (π£
2β π2)π΄2π΅2 tanhπ2β2π
β 2π2
2(π£2β πΎ2)π΄2π΅2 tanhπ2π
+ π2(π2+ 1) (π£
2β π2)π΄2π΅2 tanhπ2+2π
β ππ΄2π
1π΅2{2ππ1(2ππ1β 1) tanh2ππ1β2π β 8π2π2
1tanh2ππ1π
+2ππ1(2ππ1+ 1) tanh2ππ1+2π} = 0,
(8)
respectively. Now, splitting (7) into two real and imaginaryparts gives
π1(π1β 1) (π£
2β π2) π΅2tanhπ1β2π
β {2π2
1(π£2β π2) π΅2+ π2+ π2π 2} tanhπ1π
+ π1(π1+ 1) (π£
2β π2) π΅2 tanhπ1+2π + π tanhπ1π
+ ππ΄2tanhπ1+π2π + ππ΄2π
1tanh(2π+1)π1π = 0,
(9)
2ππ1(π£π β π
2π2) π΅ tanhπ1β1π
β 2ππ1(π£π β π
2π2) π΅ tanhπ1+1π = 0.
(10)
From (9), equating the exponents π1+ π2and π
1+ 2 gives
π2= 2, (11)
and then equating (2π + 1)π1with π
1+ 2 gives
π1=1
π. (12)
Finally, equating the exponent pairs (2π + 1)π1and π
1+ π2
gives
π2= 2ππ
1. (13)
Now the values of π1and π
2from (11) and (12) satisfy (13).
Finally, equating the coefficients of the linearly indepen-dent functions tanhπ1Β±ππ, π = Β±1, 0, Β±2 in (9) and (10) to zerogives
(π£π β π 2π2) π΅ = 0,
2 (π£2β π2) π΅2+ π2(π2+ π 2π2β π) = 0,
(π + 1) (π£2β π2) π΅2+ π2(ππ΄2+ ππ΄2π
1) = 0.
(14)
Again, equating the coefficients of the linearly independentfunctions tanhπ2Β±ππ, π = Β±1, Β±2 in (8) to zero implies
(π£2β π2)π΄2β ππ΄2π
1= 0. (15)
Solving (14)-(15) we get
π =π2π 2
π£, (16)
π΅ = πβπ β π2π 2β π2
2 (π£2β π2), (17)
π΄1= [
(π + 1) (π£2β π2) (π2β π β π
2π2) π
2 {π (π£2β π2) + ππ}
]
1/2π
, (18)
π΄2=
π (π + 1) (π2β π β π
2π 2)
2 {π (π£2β π2) + ππ}
. (19)
The relations (17), (18), and (19) introduce the restrictionsgiven by
(π β π2π 2β π2) (π£2β π2) > 0,
π (π£2β π2) (π2β π β πΎ
2π 2) {π (π£
2β π2) + ππ} > 0,
π (π£2β π2) + ππ ΜΈ= 0.
(20)
Thus the topological solution of the π and π wave functionsare given:
π (π₯, π‘) = π΄1tanh1/π [π΅ (π₯ β π£π‘)] ππ(βπ π₯+ππ‘+π),
π (π₯, π‘) = π΄2tanh2 [π΅ (π₯ β π£π‘)] .
(21)
Journal of Applied Mathematics 3
3. Bifurcation Analysis
This section will carry out the bifurcation analysis of theKlein-Gordon-Zakharov equation with power law nonlin-earity. Initially, the phase portraits will be obtained and thecorresponding qualitative analysis will be discussed. Severalinteresting properties of the solution structure will beobtained based on the parameter regimes. Subsequently, thetraveling wave solutions will be discussed from the bifurca-tion analysis.
3.1. Phase Portraits and Qualitative Analysis. We assume thatthe traveling wave solutions of (1) and (2) are of the form
π (π₯, π‘) = ππππ (π) , π (π₯, π‘) = π (π) , (22)
π = ππ₯ + ππ‘, π = ππ₯ β π£π‘, (23)
where π(π) and π(π) are real functions,π, π, π, and π£ are realconstants.
Substituting (22) and (23) into (1) and (2), we find thatπ = βπ£π/ππ
2, π and π satisfy the following system:
(π£2β π2π2) πβ (π2β π2π2β π) π + πππ + ππ
2π+1= 0,
(24)
(π£2+ π2π2) πβ ππ2(π2π)
= 0. (25)
Integrating (25) twice and letting the first integral constant bezero, we have
π =ππ2π2π
π£2β π2π2+ π, π£
2β π2π2ΜΈ= 0, (26)
where π is the second integral constant.Substituting (26) into (24), we have
(π£2β π2π2) πβ (π2β π2π2β π β ππ) π
+ (π +πππ2
π£2β π2π2)π2π+1
= 0.
(27)
To facilitate discussions, we let
πΏ =
πππ2+ π (π£
2β π2π2)
(π£2β π2π2)2
, (28)
π =π2β π2π2β π β ππ
π£2β π2π2
. (29)
Letting π = π§, then we get the following planar system:
dπdπ= π§,
dπ§dπ= βπΏπ
2π+1+ ππ.
(30)
Obviously, the above system (30) is a Hamiltonian systemwith Hamiltonian function
π»(π, π§) = π§2+
πΏ
π + 1π2π+2
β ππ2. (31)
In order to investigate the phase portrait of (30), set
π (π) = βπΏπ2π+1
+ ππ. (32)
Obviously, when πΏπ > 0, π(π) has three zero points, πβ, π0,
and π+, which are given as follows:
πβ= β(
π
πΏ)
1/2π
, π0= 0, π
+= (
π
πΏ)
1/2π
. (33)
When πΏπ β©½ 0, π(π) has only one zero point
π0= 0. (34)
Letting (ππ, 0) be one of the singular points of system (30),
then the characteristic values of the linearized system ofsystem (30) at the singular points (π
π, 0) are
πΒ±= Β±βπ
(ππ). (35)
From the qualitative theory of dynamical systems, we knowthe following.
(I) If π(ππ) > 0, (π
π, 0) is a saddle point.
(II) If π(ππ) < 0, (π
π, 0) is a center point.
(III) If π(ππ) = 0, (π
π, 0) is a degenerate saddle point.
Therefore, we obtain the bifurcation phase portraits ofsystem (30) in Figure 1.
Let
π»(π, π§) = β, (36)
where β is Hamiltonian.Next, we consider the relations between the orbits of (30)
and the Hamiltonian β.Set
ββ=π» (π+, 0)
=π» (πβ, 0)
. (37)
According to Figure 1, we get the following propositions.
Proposition 1. Suppose that πΏ > 0 and π > 0, one has thefollowing.
(I) When β β©½ βββ, system (30) does not have any closedorbits.
(II) When βββ < β < 0, system (30) has two periodic orbitsΞ1and Ξ2.
(III) When β = 0, system (30) has two homoclinic orbits Ξ3
and Ξ4.
(IV) When β > 0, system (30) has a periodic orbit Ξ5.
Proposition 2. Suppose that πΏ < 0 and π < 0, one has thefollowing.
(I) When β < 0 or β > ββ, system (30) does not have anyclosed orbits.
(II) When 0 < β < ββ, system (30) has three periodic orbitsΞ6, Ξ7, and Ξ
8.
4 Journal of Applied Mathematics
Ο
ΟΟ
Ο
ΞΈ
Ξ΄
(I)(II)
(III) (IV)
zz
z z
Ξ3
Ξ5Ξ2 Ξ4
Ξ9
Ξ6 Ξ7 Ξ12
Ξ10
Ξ8
Ξ1
Ξ11
Figure 1: The bifurcation phase portraits of system (30). (I) πΏ > 0, π > 0, (II) πΏ < 0, π β©Ύ 0, (III) πΏ < 0, π < 0, (IV) πΏ > 0, π β©½ 0.
(III) When β = 0, system (30) has two periodic orbits Ξ9and
Ξ10.
(IV) When β = ββ, system (30) has two heteroclinic orbitsΞ11and Ξ12.
Proposition 3. (I)When πΏ > 0, π β©Ύ 0 and β > 0, system (30)has a periodic orbits.
(II) When πΏ < 0, π β©½ 0, system (30) does have not anyclosed orbits.
From the qualitative theory of dynamical systems, weknow that a smooth solitary wave solution of a partialdifferential system corresponds to a smooth homoclinic orbitof a traveling wave equation. A smooth kink wave solution ora unbounded wave solution corresponds to a smooth hetero-clinic orbit of a traveling wave equation. Similarly, a periodicorbit of a traveling wave equation corresponds to a peri-odic traveling wave solution of a partial differential system.According to the above analysis, we have the following pro-positions.
Proposition 4. If πΏ > 0 and π > 0, one has the following.
(I) When βββ < β < 0, (1) and (2) have two periodic wavesolutions (corresponding to the periodic orbits Ξ
1and Ξ2
in Figure 1).(II) When β = 0, (1) and (2) have two solitary wave
solutions (corresponding to the homoclinic orbits Ξ3and
Ξ4in Figure 1).
(III) When β > 0, (1) and (2) have two periodic wavesolutions (corresponding to the periodic orbit Ξ
5in
Figure 1).
Proposition 5. If πΏ < 0 and π < 0, one has the following.
(I) When 0 < β < ββ, (1) and (2) have two periodicwave solutions (corresponding to the periodic orbit Ξ
7
in Figure 1) and two periodic blow-up wave solutions
(corresponding to the periodic orbits Ξ6and Ξ
8in
Figure 1).(II) When β = 0, (1) and (2) have periodic blow-up wave
solutions (corresponding to the periodic orbits Ξ9and
Ξ10in Figure 1).
(III) When β = ββ, (1) and (2) have two kink profile solitarywave solutions. (corresponding to the heteroclinic orbitsΞ11and Ξ12in Figure 1).
3.2. Exact TravelingWave Solutions. Firstly, wewill obtain theexplicit expressions of traveling wave solutions for (1) and (2)when πΏ > 0 and π > 0. From the phase portrait, we see thatthere are two symmetric homoclinic orbits Ξ
3and Ξ
4con-
nected at the saddle point (0, 0). In (π, π§)-plane the expres-sions of the homoclinic orbits are given as
π§ = Β±βπΏ
π + 1πββπ
2π+(π + 1) π
πΏ. (38)
Substituting (38) into dπ/dπ = π§ and integrating them alongthe orbits Ξ
3and Ξ4, we have
Β±β«
π
π1
1
π ββπ 2π+ (π + 1) π/πΏ
dπ = β πΏπ + 1
β«
π
0
dπ ,
Β±β«
π
π2
1
π ββπ 2π+ (π + 1) π/πΏ
dπ = β πΏπ + 1
β«
π
0
dπ ,
(39)
where π1= β((π + 1)π/πΏ)
1/2π and π2= ((π + 1)π/πΏ)
1/2π.Completing the above integrals we obtain
π = (β(π + 1)π
πΏsech πβππ)
1/π
π = β(β(π + 1) π
πΏsech πβππ)
1/π
.
(40)
Journal of Applied Mathematics 5
Noting (22), (23), and (26), we get the following solitary wavesolutions:
π1(π₯, π¦, π‘) = π
ππ(β
(π + 1) π
πΏsech πβππ)
1/π
,
π1(π₯, π¦, π‘) =
β (π + 1) π½π(sech πβππ)2
πΏ (π2+ π2)
+ π,
π2(π₯, π¦, π‘) = βπ
ππ(β
(π + 1) π
πΏsech πβππ)
1/π
,
π2(π₯, π¦, π‘) =
β (π + 1) π½π(sech πβππ)2
πΏ (π2+ π2)
+ π,
(41)
where πΏ is given by (28), π is given by (29), π = ππ₯ + ππ‘, andπ = ππ₯ β π£π‘.
Secondly, we will obtain the explicit expressions of trav-eling wave solutions for (1) and (2) when πΏ < 0 and π < 0.From the phase portrait, we note that there are two specialorbits Ξ
9and Ξ10, which have the same Hamiltonian with that
of the center point (0, 0). In (π, π§)-plane the expressions of theorbits are given as
π§ = Β±ββπΏ
π + 1πβπ2πβ(π + 1) π
πΏ. (42)
Substituting (42) into dπ/dπ = π§ and integrating them alongthe two orbits Ξ
9and Ξ10, it follows that
Β±β«
+β
π
1
π βπ 2πβ (π + 1) π/πΏ
dπ = ββ πΏπ + 1
β«
π
0
dπ ,
Β± β«
π
π4
1
π βπ 2πβ (π + 1) π/πΏ
dπ = ββ πΏπ + 1
β«
π
0
dπ ,
(43)
where π4= ((π + 1)π/πΏ)
1/2π.Completing the above integrals we obtain
π = Β±(β(π + 1)π
πΏcsc πββππ)
1/π
,
π = Β±(β(π + 1)π
πΏsec πββππ)
1/π
.
(44)
Noting (22), (23), and (26), we get the following periodicblow-up wave solutions:
π3(π₯, π¦, π‘) = Β±π
ππ(β
(π + 1) π
πΏcsc πββππ)
1/π
,
π3(π₯, π¦, π‘) =
β (π + 1) π½π(csc πββππ)2
πΏ (π2+ π2)
+ π,
π4(π₯, π¦, π‘) = Β±π
ππ(β
(π + 1) π
πΏsec πββππ)
1/π
,
π4(π₯, π¦, π‘) =
β (π + 1) π½π(sec πββππ)2
πΏ (π2+ π2)
+ π,
(45)
where πΏ is given by (28), π is given by (29), π = ππ₯ + ππ‘, andπ = ππ₯ β π£π‘.
4. Numerical Simulation
We decompose the function π in (1) in the form
π = π’ + ππ£ (46)
Substituting in (1) and (2) we have
π’π‘π‘β π2π’π₯π₯+ ππ’ + πππ£ + π(π’
2+ π£2)π
π’ = 0,
π£π‘π‘β π2π£π₯π₯+ ππ£ + πππ£ + π(π’
2+ π£2)π
π£ = 0,
ππ‘π‘β π2π’π₯π₯β πΌ(π’
2+ π£2)π
π₯π₯= 0.
(47)
We assume that π’ππ, π£ππ, πππis the exact solution and ππ
π, πππ,
π π
πis the approximate solution at the grid point (π₯
π, π‘π). The
proposed scheme can be displayed as
1
π2πΏ2
π‘ππ
πβπ2
β2πΏ2
π₯ππ
π+ πππ
π+ ππ π,πππ
π
+ π((ππ
π)2
+ (ππ
π)2
)π
ππ
π= 0,
1
π2πΏ2
π‘ππ
πβπ2
β2πΏ2
π₯ππ
π+ πππ
π+ ππ π,nππ
π
+ π((ππ
π)2
+ (ππ
π)2
)π
ππ
π= 0,
1
π2πΏ2
π‘π π
πβπ2
β2πΏ2
π₯π π
π+ πΌ((π
π
π)2
+ (ππ
π)2
)π
π₯π₯= 0,
(48)
where
πΏ2
π‘ππ
π= ππ+1
πβ 2ππ
π+ ππβ1
π,
πΏ2
π₯ππ
π= ππ
π+1β 2ππ
π+ ππ
πβ1.
(49)
6 Journal of Applied Mathematics
0.4
0.3
0.2
0.1
0
|q|
tx
10
5
0 β40 β200 20
40
|q(x, t)|2 topological solution
tx
10
5
0
0
β40 β200 20
40
β0.2β0.4β0.6β0.8β1
r
r(x, t) topological solution
Figure 2: Topological solution for the Klein-Gordon-Zakharovequations.
The similar notation for πΏ2π‘ππ
π, πΏ2
π₯ππ
πand πΏ2π‘π π
π, πΏ2
π₯π π
πare
πΏ2
π‘ππ
π= ππ+1
πβ 2ππ
π+ ππβ1
π,
πΏ2
π₯ππ
π= ππ
π+1β 2ππ
π+ ππ
πβ1,
πΏ2
π‘π π
π= π π+1
πβ 2π π
π+ π πβ1
π,
πΏ2
π₯π π
π= π π
π+1β 2π π
π+ π π
πβ1,
(50)
The proposed scheme is implicit and can be easily solved bythe fixed point method. The scheme is second order in spaceand time directions.
To get the numerical solution the initial conditions aretaken from the exact solution (21). Figure 2 displays thenumerical solutions of |π(π₯, π‘)| and π(π₯, π‘) at π = 1,respectively. We choose the parameter
π = 1, πΎ = 1, π = 0.5, π = 0.5, π = 0.5,
π = β0.5, π = 0.5, π = 0.5, π£ = 0.6.
(51)
5. Conclusions
This paper studied the KGZ equation in (1+1)-D with powerlaw nonlinearity from three different avenues. First, thetopological 1-soliton solution to the equation was determinedby the aid of ansatz method. The by-product of this solutionis a couple of constraint conditions that must remain valid inorder for the solitons to exist. Subsequently, the bifurcation
analysis is carried out for this equation that leads to the phaseportraits and several other solutions to the equation, using thetravelingwave hypothesis.This leads to the solitarywaves andperiodic singular waves. Finally, the numerical simulationthat was conducted using finite difference scheme leads to thesimulations for the topological soliton solutions.
These results are pretty complete in analysis. They aregoing to be extended in the future. An obvious way to expandor generalize these results is going to extend to (2 + 1)-D.These results will be reported soon. Another avenue to lookinto this equation further is to consider the perturbationterms and then obtain exact solution, and additionally studythe perturbed KGZ equation using other tools of integra-bility. These include mapping method, Lie symmetries, exp-function method, and the πΊ/πΊ-expansion method. Thesewill lead to a further plethora of solutions. Such results willbe reported in the future. That is just a foot in the door.
References
[1] G. Ebadi and A. Biswas, βApplication of the G/G-expansionmethod for nonlinear diffusion equations with nonlinearsource,β Journal of the Franklin Institute, vol. 347, no. 7, pp. 1391β1398, 2010.
[2] G. Ebadi and A. Biswas, βThe G/G method and 1-solitonsolution of the Davey-Stewartson equation,βMathematical andComputer Modelling, vol. 53, no. 5-6, pp. 694β698, 2011.
[3] D. Feng and J. Li, βDynamics and bifurcations of travellingwave solutions of R(m, n) equations,β Proceedings of the IndianAcademy of Sciences, vol. 117, no. 4, pp. 555β574, 2007.
[4] Z. Gan and J. Zhang, βInstability of standing waves for Klein-Gordon-Zakharov equations with different propagation speedsin three space dimensions,β Journal of Mathematical Analysisand Applications, vol. 307, no. 1, pp. 219β231, 2005.
[5] Z. Gan, B. Guo, and J. Zhang, βInstability of standing wave,global existence and blowup for the Klein-Gordon-Zakharovsystem with different-degree nonlinearities,β Journal of Differ-ential Equations, vol. 246, no. 10, pp. 4097β4128, 2009.
[6] M. S. Ismail and A. Biswas, β1-Soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity,βAppliedMathematics and Computation, vol. 217, no. 8, pp. 4186β4196, 2010.
[7] E. V. Krishnan and A. Biswas, βSolutions to the Zakharov-Kuznetsov equation with higher order nonlinearity bymappingand ansatz methods,β Physics of Wave Phenomena, vol. 18, no. 4,pp. 256β261, 2010.
[8] E. V. Krishnan and Y.-Z. Peng, βExact solutions to the combinedKdV-mKdV equation by the extended mapping method,β Phys-ica Scripta, vol. 73, no. 4, article 405, 2006.
[9] J. Li, βExact explicit travelling wave solutions for (π + 1)-dimensional Klein-Gordon-Zakharov equations,β Chaos, Soli-tons and Fractals, vol. 34, no. 3, pp. 867β871, 2007.
[10] L. Chen, βOrbital stability of solitary waves for the Klein-Gordon-Zakharov equations,β Acta Mathematicae ApplicataeSinica, vol. 15, no. 1, pp. 54β64, 1999.
[11] N. Masmoudi and K. Nakanishi, βFrom the Klein-Gordon-Zakharov system to a singular nonlinear SchroΜdinger system,βAnnales de lβInstitut Henri Poincare (C) Non Linear Analysis, vol.27, no. 4, pp. 1073β1096, 2010.
Journal of Applied Mathematics 7
[12] Y. Shang, Y. Huang, and W. Yuan, βNew exact traveling wavesolutions for the Klein-Gordon-Zakharov equations,β Comput-ers and Mathematics with Applications, vol. 56, no. 5, pp. 1441β1450, 2008.
[13] T. Wang, J. Chen, and L. Zhang, βConservative differencemethods for the Klein-Gordon-Zakharov equations,β Journalof Computational and Applied Mathematics, vol. 205, no. 1, pp.430β452, 2007.
[14] A. M. Wazwaz, βThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequations,β Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1505β1516, 2008.
[15] L. Zhang, βConvergence of a conservative difference scheme fora class of Klein-Gordon-SchroΜdinger equations in one spacedimension,βAppliedMathematics andComputation, vol. 163, no.1, pp. 343β355, 2005.
[16] J. Li and Z. Liu, βSmooth and non-smooth traveling wavesin a nonlinearly dispersive equation,β Applied MathematicalModelling, vol. 25, no. 1, pp. 41β56, 2000.
[17] M. Song, X. Hou, and J. Cao, βSolitary wave solutions andkink wave solutions for a generalized KDV-mKDV equation,βApplied Mathematics and Computation, vol. 217, no. 12, pp.5942β5948, 2011.
[18] M. Song and Z. Liu, βPeriodic wave solutions and their limitsfor the generalized KP-BBM equation,β Journal of AppliedMathematics, vol. 2012, Article ID 363879, 25 pages, 2012.
[19] M. Song, βApplication of bifurcation method to the generalizedZakharov equations,β Abstract and Applied Analysis, vol. 2012,Article ID 308326, 8 pages, 2012.
[20] M. Song, Z. Liu, E. Zerrad, and A. Biswas, βSingular solitonsolution and bifurcation analysis of Klein-Gordon equationwith power law nonlinearity,β Frontiers ofMathematics in China,vol. 8, no. 1, pp. 191β201, 2013.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of