research article topological soliton solution and bifurcation...

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, Girls’ 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 © 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 [1–20]. Many of theNLEEs are pretty well known in the area of theoretical physicsand applied mathematics. A few of them are the nonlinearSchrödinger’s 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 theSchrödinger-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]

𝑞𝑡𝑡− 𝑘2𝑞𝑥𝑥+ 𝑎𝑞 + 𝑏𝑟𝑞 + 𝑐

𝑞

2𝑛

𝑞 = 0, (1)

𝑟𝑡𝑡− 𝑘2𝑟𝑥𝑥= 𝑑(

𝑞

2𝑛

)𝑥𝑥, (2)

where 𝑎, 𝑏, 𝑐, 𝑑, and 𝑘 are real valued constants. Additionally𝑞(𝑥, 𝑡) is a complex valued dependent variable and 𝑟(𝑥, 𝑡) 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

𝑞 (𝑥, 𝑡) = 𝐴1tanh𝑝1𝜏𝑒𝑖𝜙, (3)

𝑟 (𝑥, 𝑡) = 𝐴2tanh𝑝2𝜏, (4)

where

𝜏 = 𝐵 (𝑥 − 𝑣𝑡) . (5)

Here, in (3) and (4)𝐴1, 𝐴2and 𝐵 are free parameters, while 𝑣

is the velocity of the soliton. The unknown exponents 𝑝1and

𝑝2will be determined, in terms of 𝑛 by the aid of balancing

principle. The phase component of (3) is given by

𝜙 = −𝜅𝑥 + 𝜔𝑡 + 𝜃, (6)

where 𝜅 represents the soliton frequency, 𝜔 is the solitonwave number, and 𝜃 is the phase constant. Substituting thehypothesis (3) and (4) into (1) and (2) yields

𝑝1(𝑝1− 1) (𝑣

2− 𝑘2) 𝐵2tanh𝑝1−2𝜏

− 2𝑖𝑝1(𝑣𝜔 − 𝜅

2𝑘2) 𝐵 tanh𝑝1−1𝜏

− {2𝑝2

1(𝑣2− 𝑘2) 𝐵2+ 𝜔2+ 𝐾2𝜅2} tanh𝑝1𝜏

+ 2𝑖𝑝1(𝑣𝜔 − 𝜅

2𝑘2) 𝐵 tanh𝑝1+1𝜏

+ 𝑝1(𝑝1+ 1) (𝑣

2− 𝑘2) 𝐵2 tanh𝑝1+2𝜏 + 𝑎 tanh𝑝1𝜏

+ 𝑏𝐴2tanh𝑝1+𝑝2𝜏 + 𝑐𝐴2𝑛

1tanh(2𝑛+1)𝑝1𝜏 = 0,

(7)

𝑝2(𝑝2− 1) (𝑣

2− 𝑘2)𝐴2𝐵2 tanh𝑝2−2𝜏

− 2𝑝2

2(𝑣2− 𝐾2)𝐴2𝐵2 tanh𝑝2𝜏

+ 𝑝2(𝑝2+ 1) (𝑣

2− 𝑘2)𝐴2𝐵2 tanh𝑝2+2𝜏

− 𝑑𝐴2𝑛

1𝐵2{2𝑛𝑝1(2𝑛𝑝1− 1) tanh2𝑛𝑝1−2𝜏 − 8𝑛2𝑝2

1tanh2𝑛𝑝1𝜏

+2𝑛𝑝1(2𝑛𝑝1+ 1) tanh2𝑛𝑝1+2𝜏} = 0,

(8)

respectively. Now, splitting (7) into two real and imaginaryparts gives

𝑝1(𝑝1− 1) (𝑣

2− 𝑘2) 𝐵2tanh𝑝1−2𝜏

− {2𝑝2

1(𝑣2− 𝑘2) 𝐵2+ 𝜔2+ 𝑘2𝜅2} tanh𝑝1𝜏

+ 𝑝1(𝑝1+ 1) (𝑣

2− 𝑘2) 𝐵2 tanh𝑝1+2𝜏 + 𝑎 tanh𝑝1𝜏

+ 𝑏𝐴2tanh𝑝1+𝑝2𝜏 + 𝑐𝐴2𝑛

1tanh(2𝑛+1)𝑝1𝜏 = 0,

(9)

2𝑖𝑝1(𝑣𝜔 − 𝜅

2𝑘2) 𝐵 tanh𝑝1−1𝜏

− 2𝑖𝑝1(𝑣𝜔 − 𝜅

2𝑘2) 𝐵 tanh𝑝1+1𝜏 = 0.

(10)

From (9), equating the exponents 𝑝1+ 𝑝2and 𝑝

1+ 2 gives

𝑝2= 2, (11)

and then equating (2𝑛 + 1)𝑝1with 𝑝

1+ 2 gives

𝑝1=1

𝑛. (12)

Finally, equating the exponent pairs (2𝑛 + 1)𝑝1and 𝑝

1+ 𝑝2

gives

𝑝2= 2𝑛𝑝

1. (13)

Now the values of 𝑝1and 𝑝

2from (11) and (12) satisfy (13).

Finally, equating the coefficients of the linearly indepen-dent functions tanh𝑝1±𝑗𝜏, 𝑗 = ±1, 0, ±2 in (9) and (10) to zerogives

(𝑣𝜔 − 𝜅2𝑘2) 𝐵 = 0,

2 (𝑣2− 𝑘2) 𝐵2+ 𝑛2(𝜔2+ 𝜅2𝑘2− 𝑎) = 0,

(𝑛 + 1) (𝑣2− 𝑘2) 𝐵2+ 𝑛2(𝑏𝐴2+ 𝑐𝐴2𝑛

1) = 0.

(14)

Again, equating the coefficients of the linearly independentfunctions tanh𝑝2±𝑗𝜏, 𝑗 = ±1, ±2 in (8) to zero implies

(𝑣2− 𝑘2)𝐴2− 𝑑𝐴2𝑛

1= 0. (15)

Solving (14)-(15) we get

𝜔 =𝑘2𝜅2

𝑣, (16)

𝐵 = 𝑛√𝑎 − 𝑘2𝜅2− 𝜔2

2 (𝑣2− 𝑘2), (17)

𝐴1= [

(𝑛 + 1) (𝑣2− 𝑘2) (𝜔2− 𝑎 − 𝜅

2𝑘2) 𝑑

2 {𝑐 (𝑣2− 𝑘2) + 𝑏𝑑}

]

1/2𝑛

, (18)

𝐴2=

𝑑 (𝑛 + 1) (𝜔2− 𝑎 − 𝑘

2𝜅2)

2 {𝑐 (𝑣2− 𝑘2) + 𝑏𝑑}

. (19)

The relations (17), (18), and (19) introduce the restrictionsgiven by

(𝑎 − 𝑘2𝜅2− 𝜔2) (𝑣2− 𝑘2) > 0,

𝑑 (𝑣2− 𝑘2) (𝜔2− 𝑎 − 𝐾

2𝜅2) {𝑐 (𝑣

2− 𝑘2) + 𝑏𝑑} > 0,

𝑐 (𝑣2− 𝑘2) + 𝑏𝑑 ̸= 0.

(20)

Thus the topological solution of the 𝑞 and 𝑟 wave functionsare given:

𝑞 (𝑥, 𝑡) = 𝐴1tanh1/𝑛 [𝐵 (𝑥 − 𝑣𝑡)] 𝑒𝑖(−𝜅𝑥+𝜔𝑡+𝜃),

𝑟 (𝑥, 𝑡) = 𝐴2tanh2 [𝐵 (𝑥 − 𝑣𝑡)] .

(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

𝑞 (𝑥, 𝑡) = 𝑒𝑖𝜂𝜑 (𝜉) , 𝑟 (𝑥, 𝑡) = 𝜓 (𝜉) , (22)

𝜂 = 𝑚𝑥 + 𝑙𝑡, 𝜉 = 𝑝𝑥 − 𝑣𝑡, (23)

where 𝜑(𝜉) and 𝜓(𝜉) are real functions,𝑚, 𝑙, 𝑝, and 𝑣 are realconstants.

Substituting (22) and (23) into (1) and (2), we find that𝑝 = −𝑣𝑙/𝑚𝑘

2, 𝜑 and 𝜙 satisfy the following system:

(𝑣2− 𝑘2𝑝2) 𝜑− (𝑙2− 𝑘2𝑚2− 𝑎) 𝜑 + 𝑏𝜑𝜓 + 𝑐𝜑

2𝑛+1= 0,

(24)

(𝑣2+ 𝑘2𝑝2) 𝜓− 𝑑𝑝2(𝜑2𝑛)

= 0. (25)

Integrating (25) twice and letting the first integral constant bezero, we have

𝜓 =𝑑𝑝2𝜑2𝑛

𝑣2− 𝑘2𝑝2+ 𝑔, 𝑣

2− 𝑘2𝑝2̸= 0, (26)

where 𝑔 is the second integral constant.Substituting (26) into (24), we have

(𝑣2− 𝑘2𝑝2) 𝜑− (𝑙2− 𝑘2𝑚2− 𝑎 − 𝑏𝑔) 𝜑

+ (𝑐 +𝑏𝑑𝑝2

𝑣2− 𝑘2𝑝2)𝜑2𝑛+1

= 0.

(27)

To facilitate discussions, we let

𝛿 =

𝑏𝑑𝑝2+ 𝑐 (𝑣

2− 𝑘2𝑝2)

(𝑣2− 𝑘2𝑝2)2

, (28)

𝜃 =𝑙2− 𝑘2𝑚2− 𝑎 − 𝑏𝑔

𝑣2− 𝑘2𝑝2

. (29)

Letting 𝜑 = 𝑧, then we get the following planar system:

d𝜑d𝜉= 𝑧,

d𝑧d𝜉= −𝛿𝜑

2𝑛+1+ 𝜃𝜑.

(30)

Obviously, the above system (30) is a Hamiltonian systemwith Hamiltonian function

𝐻(𝜑, 𝑧) = 𝑧2+

𝛿

𝑛 + 1𝜑2𝑛+2

− 𝜃𝜑2. (31)

In order to investigate the phase portrait of (30), set

𝑓 (𝜑) = −𝛿𝜑2𝑛+1

+ 𝜃𝜑. (32)

Obviously, when 𝛿𝜃 > 0, 𝑓(𝜑) has three zero points, 𝜑−, 𝜑0,

and 𝜑+, which are given as follows:

𝜑−= −(

𝜃

𝛿)

1/2𝑛

, 𝜑0= 0, 𝜑

+= (

𝜃

𝛿)

1/2𝑛

. (33)

When 𝛿𝜃 ⩽ 0, 𝑓(𝜑) has only one zero point

𝜑0= 0. (34)

Letting (𝜑𝑖, 0) be one of the singular points of system (30),

then the characteristic values of the linearized system ofsystem (30) at the singular points (𝜑

𝑖, 0) are

𝜆±= ±√𝑓

(𝜑𝑖). (35)

From the qualitative theory of dynamical systems, we knowthe following.

(I) If 𝑓(𝜑𝑖) > 0, (𝜑

𝑖, 0) is a saddle point.

(II) If 𝑓(𝜑𝑖) < 0, (𝜑

𝑖, 0) is a center point.

(III) If 𝑓(𝜑𝑖) = 0, (𝜑

𝑖, 0) is a degenerate saddle point.

Therefore, we obtain the bifurcation phase portraits ofsystem (30) in Figure 1.

Let

𝐻(𝜑, 𝑧) = ℎ, (36)

where ℎ is Hamiltonian.Next, we consider the relations between the orbits of (30)

and the Hamiltonian ℎ.Set

ℎ∗=𝐻 (𝜑+, 0)

=𝐻 (𝜑−, 0)

. (37)

According to Figure 1, we get the following propositions.

Proposition 1. Suppose that 𝛿 > 0 and 𝜃 > 0, one has thefollowing.

(I) When ℎ ⩽ −ℎ∗, system (30) does not have any closedorbits.

(II) When −ℎ∗ < ℎ < 0, system (30) has two periodic orbitsΓ1and Γ2.

(III) When ℎ = 0, system (30) has two homoclinic orbits Γ3

and Γ4.

(IV) When ℎ > 0, system (30) has a periodic orbit Γ5.

Proposition 2. Suppose that 𝛿 < 0 and 𝜃 < 0, one has thefollowing.

(I) When ℎ < 0 or ℎ > ℎ∗, system (30) does not have anyclosed orbits.

(II) When 0 < ℎ < ℎ∗, system (30) has three periodic orbitsΓ6, Γ7, and Γ

8.


ϕ

ϕϕ

ϕ

θ

δ

(I)(II)

(III) (IV)

zz

z z

Γ3

Γ5Γ2 Γ4

Γ9

Γ6 Γ7 Γ12

Γ10

Γ8

Γ1

Γ11

Figure 1: The bifurcation phase portraits of system (30). (I) 𝛿 > 0, 𝜃 > 0, (II) 𝛿 < 0, 𝜃 ⩾ 0, (III) 𝛿 < 0, 𝜃 < 0, (IV) 𝛿 > 0, 𝜃 ⩽ 0.

(III) When ℎ = 0, system (30) has two periodic orbits Γ9and

Γ10.

(IV) When ℎ = ℎ∗, system (30) has two heteroclinic orbitsΓ11and Γ12.

Proposition 3. (I)When 𝛿 > 0, 𝜃 ⩾ 0 and ℎ > 0, system (30)has a periodic orbits.

(II) When 𝛿 < 0, 𝜃 ⩽ 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 𝛿 > 0 and 𝜃 > 0, one has the following.

(I) When −ℎ∗ < ℎ < 0, (1) and (2) have two periodic wavesolutions (corresponding to the periodic orbits Γ

1and Γ2

in Figure 1).(II) When ℎ = 0, (1) and (2) have two solitary wave

solutions (corresponding to the homoclinic orbits Γ3and

Γ4in Figure 1).

(III) When ℎ > 0, (1) and (2) have two periodic wavesolutions (corresponding to the periodic orbit Γ

5in

Figure 1).

Proposition 5. If 𝛿 < 0 and 𝜃 < 0, one has the following.

(I) When 0 < ℎ < ℎ∗, (1) and (2) have two periodicwave solutions (corresponding to the periodic orbit Γ

7

in Figure 1) and two periodic blow-up wave solutions

(corresponding to the periodic orbits Γ6and Γ

8in

Figure 1).(II) When ℎ = 0, (1) and (2) have periodic blow-up wave

solutions (corresponding to the periodic orbits Γ9and

Γ10in Figure 1).

(III) When ℎ = ℎ∗, (1) and (2) have two kink profile solitarywave solutions. (corresponding to the heteroclinic orbitsΓ11and Γ12in Figure 1).

3.2. Exact TravelingWave Solutions. Firstly, wewill obtain theexplicit expressions of traveling wave solutions for (1) and (2)when 𝛿 > 0 and 𝜃 > 0. From the phase portrait, we see thatthere are two symmetric homoclinic orbits Γ

3and Γ

4con-

nected at the saddle point (0, 0). In (𝜑, 𝑧)-plane the expres-sions of the homoclinic orbits are given as

𝑧 = ±√𝛿

𝑛 + 1𝜑√−𝜑

2𝑛+(𝑛 + 1) 𝜃

𝛿. (38)

Substituting (38) into d𝜑/d𝜉 = 𝑧 and integrating them alongthe orbits Γ

3and Γ4, we have

±∫

𝜑

𝜑1

1

𝑠√−𝑠2𝑛+ (𝑛 + 1) 𝜃/𝛿

d𝑠 = √ 𝛿𝑛 + 1

∫

𝜉

0

d𝑠,

±∫

𝜑

𝜑2

1

𝑠√−𝑠2𝑛+ (𝑛 + 1) 𝜃/𝛿

d𝑠 = √ 𝛿𝑛 + 1

∫

𝜉

0

d𝑠,

(39)

where 𝜑1= −((𝑛 + 1)𝜃/𝛿)

1/2𝑛 and 𝜑2= ((𝑛 + 1)𝜃/𝛿)

1/2𝑛.Completing the above integrals we obtain

𝜑 = (√(𝑛 + 1)𝜃

𝛿sech 𝑛√𝜃𝜉)

1/𝑛

𝜑 = −(√(𝑛 + 1) 𝜃


1/𝑛

.

(40)


Noting (22), (23), and (26), we get the following solitary wavesolutions:

𝑞1(𝑥, 𝑦, 𝑡) = 𝑒

𝑖𝜂(√

(𝑛 + 1) 𝜃


1/𝑛

,

𝑟1(𝑥, 𝑦, 𝑡) =

− (𝑛 + 1) 𝛽𝜃(sech 𝑛√𝜃𝜉)2

𝛿 (𝑝2+ 𝑚2)

+ 𝑔,

𝑞2(𝑥, 𝑦, 𝑡) = −𝑒

𝑖𝜂(√

(𝑛 + 1) 𝜃


1/𝑛

,

𝑟2(𝑥, 𝑦, 𝑡) =

− (𝑛 + 1) 𝛽𝜃(sech 𝑛√𝜃𝜉)2

𝛿 (𝑝2+ 𝑚2)

+ 𝑔,

(41)

where 𝛿 is given by (28), 𝜃 is given by (29), 𝜂 = 𝑚𝑥 + 𝑙𝑡, and𝜉 = 𝑝𝑥 − 𝑣𝑡.

Secondly, we will obtain the explicit expressions of trav-eling wave solutions for (1) and (2) when 𝛿 < 0 and 𝜃 < 0.From the phase portrait, we note that there are two specialorbits Γ

9and Γ10, which have the same Hamiltonian with that

of the center point (0, 0). In (𝜑, 𝑧)-plane the expressions of theorbits are given as

𝑧 = ±√−𝛿

𝑛 + 1𝜑√𝜑2𝑛−(𝑛 + 1) 𝜃

𝛿. (42)

Substituting (42) into d𝜑/d𝜉 = 𝑧 and integrating them alongthe two orbits Γ

9and Γ10, it follows that

±∫

+∞

𝜑

1

𝑠√𝑠2𝑛− (𝑛 + 1) 𝜃/𝛿

d𝑠 = √− 𝛿𝑛 + 1

∫

𝜉

0

d𝑠,

± ∫

𝜑

𝜑4

1

𝑠√𝑠2𝑛− (𝑛 + 1) 𝜃/𝛿

d𝑠 = √− 𝛿𝑛 + 1

∫

𝜉

0

d𝑠,

(43)

where 𝜑4= ((𝑛 + 1)𝜃/𝛿)

1/2𝑛.Completing the above integrals we obtain

𝜑 = ±(√(𝑛 + 1)𝜃

𝛿csc 𝑛√−𝜃𝜉)

1/𝑛

,

𝜑 = ±(√(𝑛 + 1)𝜃

𝛿sec 𝑛√−𝜃𝜉)

1/𝑛

.

(44)

Noting (22), (23), and (26), we get the following periodicblow-up wave solutions:

𝑞3(𝑥, 𝑦, 𝑡) = ±𝑒

𝑖𝜂(√

(𝑛 + 1) 𝜃

𝛿csc 𝑛√−𝜃𝜉)

1/𝑛

,

𝑟3(𝑥, 𝑦, 𝑡) =

− (𝑛 + 1) 𝛽𝜃(csc 𝑛√−𝜃𝜉)2

𝛿 (𝑝2+ 𝑚2)

+ 𝑔,

𝑞4(𝑥, 𝑦, 𝑡) = ±𝑒

𝑖𝜂(√

(𝑛 + 1) 𝜃

𝛿sec 𝑛√−𝜃𝜉)

1/𝑛

,

𝑟4(𝑥, 𝑦, 𝑡) =

− (𝑛 + 1) 𝛽𝜃(sec 𝑛√−𝜃𝜉)2

𝛿 (𝑝2+ 𝑚2)

+ 𝑔,

(45)

where 𝛿 is given by (28), 𝜃 is given by (29), 𝜂 = 𝑚𝑥 + 𝑙𝑡, and𝜉 = 𝑝𝑥 − 𝑣𝑡.

4. Numerical Simulation

We decompose the function 𝑞 in (1) in the form

𝑞 = 𝑢 + 𝑖𝑣 (46)

Substituting in (1) and (2) we have

𝑢𝑡𝑡− 𝑘2𝑢𝑥𝑥+ 𝑎𝑢 + 𝑏𝑟𝑣 + 𝑐(𝑢

2+ 𝑣2)𝑛

𝑢 = 0,

𝑣𝑡𝑡− 𝑘2𝑣𝑥𝑥+ 𝑎𝑣 + 𝑏𝑟𝑣 + 𝑐(𝑢

2+ 𝑣2)𝑛

𝑣 = 0,

𝑟𝑡𝑡− 𝑘2𝑢𝑥𝑥− 𝛼(𝑢

2+ 𝑣2)𝑛

𝑥𝑥= 0.

(47)

We assume that 𝑢𝑛𝑚, 𝑣𝑛𝑚, 𝑟𝑛𝑚is the exact solution and 𝑈𝑛

𝑚, 𝑉𝑛𝑚,

𝑅𝑛

𝑚is the approximate solution at the grid point (𝑥

𝑚, 𝑡𝑛). The

proposed scheme can be displayed as

1

𝑘2𝛿2

𝑡𝑈𝑛

𝑚−𝑘2

ℎ2𝛿2

𝑥𝑈𝑛

𝑚+ 𝑎𝑈𝑛

𝑚+ 𝑏𝑅𝑚,𝑛𝑈𝑛

𝑚

+ 𝑐((𝑈𝑛

𝑚)2

+ (𝑈𝑛

𝑚)2

)𝑛

𝑈𝑛

𝑚= 0,

1

𝑘2𝛿2

𝑡𝑉𝑛

𝑚−𝑘2

ℎ2𝛿2

𝑥𝑉𝑛

𝑚+ 𝑎𝑉𝑛

𝑚+ 𝑏𝑅𝑚,n𝑉𝑛

𝑚

+ 𝑐((𝑈𝑛

𝑚)2

+ (𝑉𝑛

𝑚)2

)𝑛

𝑉𝑛

𝑚= 0,

1

𝑘2𝛿2

𝑡𝑅𝑛

𝑚−𝑘2

ℎ2𝛿2

𝑥𝑅𝑛

𝑚+ 𝛼((𝑈

𝑛

𝑚)2

+ (𝑉𝑛

𝑚)2

)𝑛

𝑥𝑥= 0,

(48)

where

𝛿2

𝑡𝑈𝑛

𝑚= 𝑈𝑛+1

𝑚− 2𝑈𝑛

𝑚+ 𝑈𝑛−1

𝑚,

𝛿2

𝑥𝑈𝑛

𝑚= 𝑈𝑛

𝑚+1− 2𝑈𝑛

𝑚+ 𝑈𝑛

𝑚−1.

(49)


0.4

0.3

0.2

0.1

0

|q|

tx

10

5

0 −40 −200 20

40

|q(x, t)|2 topological solution

tx

10

5

0

0

−40 −200 20

40

−0.2−0.4−0.6−0.8−1

r

r(x, t) topological solution

Figure 2: Topological solution for the Klein-Gordon-Zakharovequations.

The similar notation for 𝛿2𝑡𝑉𝑛

𝑚, 𝛿2

𝑥𝑉𝑛

𝑚and 𝛿2𝑡𝑅𝑛

𝑚, 𝛿2

𝑥𝑅𝑛

𝑚are

𝛿2

𝑡𝑉𝑛

𝑚= 𝑉𝑛+1

𝑚− 2𝑉𝑛

𝑚+ 𝑉𝑛−1

𝑚,

𝛿2

𝑥𝑉𝑛

𝑚= 𝑉𝑛

𝑚+1− 2𝑉𝑛

𝑚+ 𝑉𝑛

𝑚−1,

𝛿2

𝑡𝑅𝑛

𝑚= 𝑅𝑛+1

𝑚− 2𝑅𝑛

𝑚+ 𝑅𝑛−1

𝑚,

𝛿2

𝑥𝑅𝑛

𝑚= 𝑅𝑛

𝑚+1− 2𝑅𝑛

𝑚+ 𝑅𝑛

𝑚−1,

(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 |𝑞(𝑥, 𝑡)| and 𝑟(𝑥, 𝑡) at 𝑛 = 1,respectively. We choose the parameter

𝑛 = 1, 𝐾 = 1, 𝜅 = 0.5, 𝑎 = 0.5, 𝑏 = 0.5,

𝑐 = −0.5, 𝑑 = 0.5, 𝑐 = 0.5, 𝑣 = 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 𝐺/𝐺-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.

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