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Seadawy Aly R., Lu Dian-Chen, Arshad Muhammad. Stability Analysis of Solitary Wave Solutions for Coupled and (2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications. Communications in Theoretical Physics, 2018, 69(6): 676  
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Stability Analysis of Solitary Wave Solutions for Coupled and (2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications
Seadawy Aly R.1, 2, *, Lu Dian-Chen3, †, Arshad Muhammad3, ‡
Mathematics Department, Faculty of Science, Taibah University, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
Mathematics Department, Faculty of Science, Beni-Suef University, Egypt
Faculty of Science, Jiangsu University, Zhenjiang 212013, China

 

† Corresponding author. E-mail: Aly742001@yahoo.com dclu@ujs.edu.cn 5103150316@ujs.edu.cn

Abstract
Abstract

The searching exact solutions in the solitary wave form of non-linear partial differential equations (PDEs) play a significant role to understand the internal mechanism of complex physical phenomena. In this paper, we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the (2+1)-dimensional cubic Klein-Gordon (K-G) equation. The Klein-Gordon equations are relativistic version of Schrödinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which several solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions of PDEs arise in mathematical physics.

Keyword:modified extended mapping method;coupled Klein-Gordon equation;cubic Klein-Gordon equation;solitons;solitary wave solutions;periodic solutions
1 Introduction

Nonlinear partial differential equations exist in all fields of engineering and science, such as optical fibers, fluid mechanics, plasma physics, chemical kinematics, biology, chemical physics etc. and also utilized to describe the complex aspects in these areas. Thus, the study to search for exact solutions of nonlinear PDEs is extremely crucial. Therefore, to search efficient techniques to construct analytic of non-linear PDEs have pinched a plenty of curiosity via a diverse group of scientists and researchers. Several symbolic techniques have been developed by different researchers and employed to get the exact solutions of nonlinear PDEs in different form, for example; Bäcklund transformations,[1] variational method,[2] Darboux transformation,[3] modified direct algebraic method,[4–5] auxiliary and simple equation methods,[6–7] Hirotas bilinear method,[8] trial equation technique,[9] inverse scattering scheme,[10] the extended tanh method, the generalized Kudryashov method,[11] (G′/G)-expansion method,[12] mapping method,[13] expansion method,[14] and many more.[15–19] In general, there are numerous researchers organized in the area of non-linear science.[20–21] Solitons solutions got more attention of researchers about the study interactions, structures, and more properties.[22–28]

The mean of the current study is to utilize the powerful proposed mapping method for the Klein-Gordon equations. As a result, novel exact solutions in more generalized and different form are obtained. Modulation instability analysis is employed to argue discuss the stability of solitary are solutions in both the normal dispersion and the anomalous regime.

The coupled Klein-Gordon equations:[29]

The cubic Klein-Gordon equation[30] is as

here, the constants δ and β are non-zero. The coupled K-G equation (1) is a relativistic interpretation of the Schrödinger equation. It is utilized to express the relation of relativistic energy-momentum in the form of quantized version. The cubic Klein-Gordon equation (2) is utilized to form several different non-linear phenomena, as well as the promulgation of crystals dislocation, the action of elementary particles and the proliferation of fluxions in Josephson junctions.

The rest of paper is ordered as follows. In Sec. 1, as an introduction is given. The important steps of proposed modified extended mapping scheme are given in Sec. 2. The applications of the proposed method on Eqs. (1) and (2) are revealed in Sec. 3. The modulation instability analysis is discussed in Sec. 4. The result and discussion are revealed in Sec. 5. The conclusion is given in Sec. 6.

2 Description of Modified Extended Mapping Method

In this section, we will present the algorithm of modified extended mapping method for nonlinear partial differential equations (PDEs). Let us assume a non-linear PDE in general form having two independent variable x and t as

the function u is unknown of x and t, and the function H is a polynomial with respect to u or prescribed variables, which contain both linear and non-linear terms of highest order derivatives of unknown function and can be reduced via employing transformation to a polynomial function in which the all real variable can be amalgamate into a complex variable. The key steps are as:

3 Application of Modified Extended Mapping Method
3.1 Coupled Klein-Gordon Equations

Performing the traveling wave transformation u(x,t) = U(ζ), v(x,t) = V(ζ) and ζ = k (x − ωt) on Eq. (1), we get

Applying homogeneous balance principle on Eq. (6), we assume the solution of Eq. (6) is as

Substituting Eq. (8) into Eq. (6) and setting the coefficients of ϕiϕ(i) to zero, we get a system of equations in parameters a0, a1, b1, c1, a2, b2, c2, α0, α1, . . ., α6, k and ω. Mathematica software is utilized to solve the obtaining system of equations. We obtain the following families of solutions:

The following traveling wave solutions of Eq. (1) are obtained from Eq. (9) as:

where , , .

One can construct more solitary wave solutions of Eq. (1) from Set 2 in the same way.

We obtained solitary wave solutions of Eq. (1) from Set 1 as follows:

One can get further solutions in solitary wave form of Eq. (1) from Set 2 in the same way.

We obtain solitary wave solutions of Eq. (1) from Set 2 as follows:

One can achieve more solitary wave form solutions of Eq. (1) from other sets in similar way.

3.2 The (2+1)-Dimensional Cubic Klein-Gordon Equation

Performing the transformation u(x,y,t) = U(ζ) and ζ = k1x + k2y − ωt on Eq. (2), we get

Applying homogeneous balance principle on Eq. (34), we assume the solution of Eq. (34) is as

Substituting Eq. (35) along with Eq. (5) into Eq. (34) and setting the coefficients of ϕiϕ(i) to zero, we got a system of equations in a0,a1,b1,c1,a2,b2,c2,α0,α1, . . ., α6, δ, β, k and ω. Mathematica software is utilized to solve the obtaining system of equations. We obtain the following families of solutions:

We construct solitary wave solutions of Eq. (2) from Set 1 as follows:

One can obtain more new exact solutions of Eq. (2) from Sets 2–5 in similar way.

The following traveling wave solutions of Eq. (2) are obtained from Set 1 as:

We can also obtain more new solitary wave solutions of Eq. (2) from Set 2 in similar way.

We construct solitary wave solutions of Eq. (2) from Set 1 as follows:

In the similar, one can achieve more solitary wave solutions of Eq. (2) from Set 2.

4 Modulation Instability

Several non-linear evolution equations of higher order illustrating an instability that directs to examine the modulation of the steady state as a results of interaction among the dispersive and non-linear effects. The modulation instability of coupled K-G equations and cubic K-G equation are studied by utilizing linear stability analysis (LSA).[31–33]

4.1 Coupled Klein-Gordon Equations

The study state solution of coupled Klein-Gordon equation has the form

here, the optical power P is the normalized. The perturbation ψi(x,t) (i = 1,2) is investigated via using LSA. Putting Eq. (62) into Eq. (1) and linearizing, yields

Assume the solution of Eq. (63) in the form

where, the wave number k and frequency ω of perturbation are the normalized. The dispersion relation ω = ω(k) of a constant coefficient linear evolution equation enumerates how time oscillations ekx are associated to spatial oscillations eωt of k, putting Eq. (64) in Eq. (63), the ω = ω(k) in following form is obtained as

where

The dispersion relation Eq. (65) reveals the steady-state stability depends on the self-phase modulation, stimulated Raman scattering, group velocity dispersion and wave number. If algebraic expression H ≥ 0, i.e. the ω is real for all k, then the steady state is stable against small perturbations. On the other hand, the steady-state solution becomes unstable if H < 0, i.e. the ω is imaginary part since the perturbation grows exponentially. One can easily see that, for the occurrence of modulation stability when H < 0. In this condition, the growth rate of modulation stability gain spectrum h(k) could be revealed as

4.2 The (2+1)-Dimensional Cubic Klein-Gordon Equation

The study state solution of (2+1)-dimensional cubic Klein Gordon equation has the form

where, the optical power P is the normalized. The perturbation ψi(x,t) (i = 1, 2) is investigated via using LSA. Putting Eq. (62) into Eq. (2) and linearizing, yields

Assume the solution of Eq. (68) in the form

where the wave numbers k1, k2 and frequency ω of perturbation are the normalized. Substituting Eq. (69) in Eq. (68), the dispersion relation is obtained in the following form as

The dispersion relation equation (70) reveals the steady-state stability depends on the self-phase modulation, stimulated Raman scattering, group velocity dispersion and wave number. If algebraic expression , i.e. the ω is real for all k1 and k2, then the steady state is stable against small perturbations. On the other hand, the steady-state solution becomes unstable if , i.e. the ω is imaginary part since the perturbation grows exponentially. One can easily see that, for the occurrence of modulation stability when . In this condition, the growth rate of modulation stability gain spectrum h(k1, k2) could be revealed as

5 Results and Discussion

In this section, we discuss the obtained exact solutions and analyse it with the existing in the literature.

Firstly, for the coupled Klein-Gordon equations; the authors in Ref. [29] used the modified simple equation method (MSEM) and achieved some exact solutions in solitary wave form with functions of trigonometric and hyperbolic structure. The obtained solutions (26) and (27) are similar to solutions (3.44) and (3.42) of Ref. [29]. The solutions (32) and (33) are similar to solutions (3.43) and (3.41) of Ref. [29]. The remaining of our constructed solutions of this equation are novel and have not formulated before.

Secondly, for the cubic Klein-Gordon equation; the authors in Ref. [30] also employed the MSEM to this equation and achieved some exact solutions. The obtained solutions (48) and (60) are similar to solutions (3.18) and (3.19) of Ref. [30]. The remaining of our constructed solutions of this equation are novel and have not formulated before.

Analyzing our solutions with the above stated solutions for both models, we determined that our exact solutions are newly constructed solutions with some important physical meaning, for instance; the tangent hyperbolic occurs in the cal-culation of magnetic moment and immediateness of special relativity, the secant hyperbolic occurs in the laminar jet profile and the sine hyperbolic occurs in the gravitational potential of a cylinder and the calculation of Roche limit.[34]

In Fig. 1, Figs. 1(a) and 1(b) evaluate dark solitary wave and bright Solitary wave solutions of solution (11) at α2 = 1.5, α3 = 1.5, α4 = −1, k = 0.5, and Figs. 1(c) and 1(d) denote the solitary wave solutions in different form of solution (13) at α2 = 5, α3 = 1.5, α4 = 1, k = 0.5. Figures 2(a) and 2(b) evaluate Kink Solitary wave and anti-Kink solitary wave solutions of solution (15) respectively at α2 = 1, α3 = 2, α4 = 1, k = 0.5, and Figs. 2(c) and 2(d) evaluate periodic Solitary wave solutions in different form of solution (20) at α2 = 1.5, α4 = 1, ω = 1.5.

Fig. 1 Exact solutions in various shapes are plotted of Family 1 solutions.
Fig. 2 Exact Solitary wave in different shapes are plotted of solutions.

In Fig. 3, Figs. 3(a) and 3(b) evaluate periodic Solitary wave and solitary wave solutions of solutions (39) and (40) respectively at α2 = −1.5, α4 = 1, k1 = 1.5, k2 = 1, δ = 1.5, β = 1. Figures 4(a) and 4(b) evaluate Solitary wave and periodic solitary wave in different form of solutions (48) and (49) at α2 = −1.5, α4 = 1, k1 = 1.5, ω = 1, δ = 0.5, β = 1 and α2 = 1.5, α4 = 1, k1 = 1.5, ω = 1, δ = 0.5, β = 1 respectively, and Figs. 4(c) and 4(d) evaluate dark Solitary wave and bright solitary wave of solutions (52) and (53) respectively at α2 = 1.5, α3 = 1.5, α4 = −1, k1 = 1, ω = 1.5, δ = 0.5, β = 1. The dispersion relation (65) among frequency (ω) and wave numbers (k) is shown in Fig. 5(a) and the dispersion relation (70) among frequency (ω) and wave numbers (k1, k2) are shown in Fig. 5(b).

Fig. 3 Exact Solitary wave in different shapes are plotted of Family 1 solutions.
Fig. 4 Exact solitary wave in various shapes are plotted.
Fig. 5 Graph of dispersion relation of Eqs. (65) and (70).
6 Conclusion

In this article, we successfully implemented the powerful proposed modified extended mapping method to achieve the solitary wave solutions of the coupled K-G and cubic K-G equations. The obtained solutions are more general and in different forms such as the solitary waves in the form of bright and dark, periodic, hyperbolic etc. We compared our solutions with the existing solutions to these two models and claimed that many solutions are novel. The equations admit the enormous diversity of possible solutions for only values of a small subset of parameters, which helps to understand the physical phenomena’s of this equation. The moments of some solutions graphically and the formation conditions for dark and bright solitons were obtained. An analytic expression for the modulation instability has been established by utilizing modulation instability which confirms that all exact solutions are stable. The efficiency and simplicity of the proposed modified extended mapping method show that it can be used to various types of different nonlinear models that arise in the various areas of nonlinear science.

Reference
[1] Anderson Ian M. Fels Mark E. J. Geom. Phys. 102 2016 1
[2] Seadawy A. R. Eur. Phys. J. Plus 130 2015 182
[3] Zhao Qian Wu Lihua Lin F. Appl. Math. Lett. 67 2017 1
[4] Arshad M. Aly Seadawy, Dianchen Lu, and Jun Wang, Results in Physics 6 2016 1136
[5] Arshad M. Seadawy A. R. Lu D. Wang J. Chin. J. Phys. 55 2017 780
[6] Tariq Kalim Ul-Haq Seadawy A. R. Results in Physics 7 2017 1143
[7] Arshad M. Seadawy Aly R. Lu Dianchen Optik 128 2017 40
[8] Jia T. T. Chai Y. Z. Hao H. Q. Superlattice Microst. 105 2017 172
[9] Triki H. Wazwaz A. M. Phys. Lett. A 380 2016 1260
[10] Lassas M. Mueller J. L. Siltanen S. Stahel A. Physica D 241 2012 1322
[11] Baskonus H. M. Bulut H. Entropy 17 2015 4255
[12] Song M. Ge Y. Comput. Math. Appl. 60 2010 1220
[13] Seadawy A. R. Math. Method Appl. Sci. 40 2017 1598
[14] Syam Kumar A. M. Prasanth J. P. Bannur Vishnu M. Physica A 432 2015 71
[15] Zhai J. Zheng B. J. Math. Anal. Appl. 445 2017 81
[16] Sinha D. Ghosh P. K. Phys. Lett. A 381 2017 124
[17] Min Li Xu Tao Lei Wang Nonlinear Dyn. 80 2015 1451
[18] Seadawy A. R Appl. Math. Inf. Sci. 10 2016 209
[19] Seadawy A. R. El-Rashidy K. Math. Comput. Model. 57 2013 1371
[20] Seadawy Aly Eur. Phys. J. Plus 132 2017 1
[21] Atangana A. Alqahtani R. T. Open Physics 14 2016 106
[22] Alam Md. N. Belgacem F. B. M. Int. J. Mod. Nonlinear Theory Appl. 5 2016 28
[23] Seadawy Aly R. Inter. J. Computat. Meth. 15 2018 1
[24] Lu Dianchen Seadawy Aly Arshad M. Optical and Quantum Electronics 50 2018 1
[25] Seadawy Aly J. Electromagnet. Waves 31 2017 1353
[26] Seadawy Aly R. European Physical Journal Plus 132 2017 29 1:13
[27] Seadawy Aly R. Pramana - J. Phys. 89 2017 1
[28] Seadawy Aly Optik 139 2017 31
[29] Khan K. Akbar M. A. Islam S. M. R. Springer Plus 3 2014 724
[30] Khan K. Ali Akbar M. J. of the Association of Arab Universities for Basic and Applied Sciences 15 2014 74
[31] Agrawal G. P. Nonlinear Fiber Optics 5 New York 2013
[32] Arshad M. Seadawy A. R. Lu D. Eur. Phys. J. Plus 132 2017 1
[33] Seadawy Aly R. Arshad M. Lu Dianchen Eur. Phys. J. Plus 132 2017 1
[34] Beyer W. H. CRC Standard Mathematical Tables 28 Academic Press Boca Raton 1987