SciELO - Scientific Electronic Library Online

 
vol.65 issue1A SU(5)×Z2 kink solution and its local stabilityAnalytical solution of the time fractional diffusion equation and fractional convection-diffusion equation author indexsubject indexsearch form
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • Have no similar articlesSimilars in SciELO

Share


Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.65 n.1 México Jan./Feb. 2019  Epub Nov 09, 2019

 

Investigación

Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity

Behzad Ghanbaria 

J.F. Gómez-Aguilarb 

aDepartment of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran.

bCONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, 62490, Cuernavaca Morelos, México. e-mail: jgomez@cenidet.edu.mx

Abstract

By using the generalized exponential rational function method, we obtain new periodic and hyperbolic soliton solutions for the conformable Ginzburg-Landau equation with the Kerr law nonlinearity. The conformable derivative is considered to obtain the exact solutions under constraint conditions. Numerical simulations are performed to confirm the efficiency of the proposed method.

Keywords: Soliton solutions; generalized exponential rational function method; Ginzburg Landau equation; conformable time fractional derivative; Kerr law nonlinearity

PACS: 02.30.Jr; 03.65.Fd; 04.20.Jb

1. Introduction

The Ginzburg-Landau equation describes the optical soliton propagation through a wide range of waveguides such as crystals, optical metamaterials, optical fibers, and optical couplers [1-9]. Many powerful methods have been established to find soliton solutions of the Ginzburg-Landau equation including the modified simple equation method [10,11], the semi-inverse variational principle [12,13], the extended Jacobi elliptic function expansion method [14,15], the exponential rational function [16], the generalized exponential rational function method (GERFM) [17], among others.

Due to the complex nature of the optical soliton propagation, several works consider the fractional calculus to construct new optical soliton solutions [18-21]. Nevertheless, fractional derivatives do not obey some basic properties of integer derivative such as product rule and chain rule. Recently, a local derivative called conformable derivative has been formulated by Khalil in [22]. The conformable calculus satisfies all the properties of the standard calculus, for instance, the chain rule. This derivative can be considered to be a natural extension of the classical derivative [23-30].

Let f : [0,∞) → ℝ, the conformable derivative of a function f(t) of order α, is defined as [22]

Dtαf(t)=limϵ0f(t+ϵt1-α)-f(t)ϵ,    

α(0,1],t>0. (1)

In this paper, the conformable GERFM is employed to study the complex time Ginzburg-Landau equation with Kerr law nonlinearity [31].

iDtαq+aqxx+b|q|2q=β[2|q|2(|q|2)xx-[(|q|2)x]2]+γq|q|2q*, (2)

where Dtα is the conformable derivative of order α ∈ (0,1], x represents the non-dimensional distance across the fiber, a represents the coefficient of group velocity dispersion, and b represents the coefficient of nonlinearity. The term β arises from the perturbation effects and γ is related to the detuning effect. All above-mentioned parameters are constants real values.

2. Overview of the generalized exponential rational function method

Let us state the main steps of GERFM as follows [17]

1. Let us take into account the following nonlinear partialdifferential equation

L(Υ,Υx,Υt,Υxx,)=0. (3)

Using the transformations Υ = Υ(χ) and χ = σxϕt, in nonlinear partial differential equation (3), we define attain

L(Υ,Υ',Υ,)=0, (4)

which is proposed as an ordinary differential equation; where the values of σ and ϕ will be determined later.

2. Consider Eq. (4) has the solution of the form

Υ(χ)=A0+k=1MAkΨ(χ)k+k=1MBkΨ(χ)-k, (5)

where

Ψ(χ)=p1eq1χ+p2eq2χp3eq3χ+p4eq4χ. (6)

The values of constants p i ,q i (1 ≤ i ≤ 4), A 0 ,A k and B k (1 ≤ kM) are determined, in such a way that solution (5) always satisfy Eq. (4). By considering the homogenous balance principle, the value of M is determined.

3. Placing Eq. (5) into Eq. (4), we give the following algebraic equation Ξ(Λ1 ,Λ2 ,Λ3 ,Λ4) = 0, in terms of Λ i = e qiχ for i = 1,...,4. After setting each of the coefficients of variables in Ξ to zero, a system of nonlinear equations in terms of p i ,q i (1 ≤ i ≤ 4), and σ,ϕ,A 0 ,A k and B k (1 ≤ kM) are generated.

4. By solving the above equations systems using any symbolic computation software, the values of p i ,q i (1 ≤ i ≤ 4), A 0 ,A k , and B k (1 ≤ kM) are determined, replacing these values into Eq. (5) provides us the soliton solutions of Eq. (3).

3. Application

In order to find solutions of Eq. (2), the following new variables are introduced

q(x,t)=Θ(χ)eiΦ(x,t),

ξ=x-(να)tα,

Φ(x,t)=-kx+(wα)tα, (7)

where ν and k are the speed and frequency of the soliton, respectively; w represents the wave number of the soliton. Considering Eq. (7), we convert Eq. (2) in the following expression

(a-4β)Θ-(w+ak2+γ)Θ+bΘ3=0, (8)

from real part

ν=-2ak, (9)

and Eq. (9), from the imaginary parts.

If we apply the balance principle on the terms Θ3 and Θ” in Eq. (8), we have 3M = M + 2, so M = 1. Using Eq. (6) together with M = 1, we have

Θ(χ)=A0+A1Ψ(χ)+B1Ψ(χ). (10)

Proceeding as outlined in second section, we obtain the following sets of solutions

Set 1:

One obtains p = [−1,0,1,1] and q = [0,0,1,1], so Eq. (6) turns to

Ψ(χ)=-11+eχ. (11)

We also obtain

k=k, w=-k2a-a/2+2β-γ,    

A0=-24β-a2b,

A1=-24β-ab,B1=0.

Placing values in Eqs. (10) and (11), yields the following solution

Θ(χ)=(1-eχ)4β-a2b(2+2eχ)

and

q1(x,t)=((1-e(x+2akαtα))4β-a2b(2+2e(x+2akαtα)))×ei(-kx+(wα)tα). (12)

Set 2:

One obtains p = [−3,−2,1,1] and q = [1,0,1,0], so Eq. (6) turns to

Ψ(χ)=-3-2eχ1+eχ. (13)

We also obtain

k=24β-a-2γ-2w2a, w=w,  

A0=-524β-a2b,A1=0,

B1=-624β-ab.

Placing values in Eqs. (10) and (13), yields the following solution

Θ(χ)=24β-a(2eχ-3)b(6+4eχ)

and

q2(x,t)=(24β-a(2e(x+2akαtα)-3)b(6+4e(x+2akαtα)))×ei(-kx+(wα)tα). (14)

Set 3: One obtains p = [2,0,1,1] and q = [−1,0,1,−1], so Eq. (6) turns to

Ψ(χ)=cosh(χ)-sinh(χ)cosh(χ). (15)

We also obtain

k=8β-2a-γ-wa, w=w,

A0=24β-ab,   

A1=-24β-ab, B1=0. (16)

Placing values in Eqs. (10) and (15), yields the following solution

Θ(χ)=sinh(χ)24β-abcosh(χ)

and

q3(x,t)=(sinh(x+2akαtα)24β-abcosh(x+2akαtα))×ei(-kx+(wα)tα). (17)

Set 4:

One obtains p = [−3,−1,1,1] and q = [1,−1,1,−1], so Eq. (6) turns to

Ψ(χ)=-sinh(χ)+2cosh(χ)cosh(χ). (18)

We also obtain

k=8β-2a-γ-+wa, w=w,  

A0=224β-ab,

A1=0,          B1=324β-ab.

Placing values in Eqs. (10) and (18), yields the following solution

Θ(χ)=24β-a(2sinh(χ)+cosh(χ))b(2cosh(χ)+sinh(χ)),

and

q4(x,t)=(24β-a(2sinh(x+2akαtα)+cosh(x+2akαtα))b(2cosh(x+2akαtα)+sinh(x+2akαtα)))×ei(-kx+(wα)tα). (19)

Set 5:

One obtains p = [1 − i,−1 − i,−1,1] and q = [i,i,i,i], so Eq. (6) turns to

Ψ(χ)=cos(χ)-sin(χ)sin(χ). (20)

We also obtain

k=k,          w=-k2a+2a-8β-γ,          A0=24β-ab,          A1=0,          B1=224β-ab.

Placing values in Eqs. (10) and (20), yields the following solution

Θ(χ)=(sin(χ)+cos(χ))24β-ab(cos(χ)-sin(χ))

and

q5(x,t)=((sin(x+2akαtα)+cos(x+2akαtα))24β-ab(cos(x+2akαtα)-sin(x+2akαtα)))×ei(-kx+(wα)tα). (21)

Set 6:

One obtains p = [−2 − i,2 − i,−1,1] and q = [i,i,i,i], so Eq. (6) turns to

Ψ(χ)=2sin(χ)+cos(χ)sin(χ). (22)

We also obtain

k=k,           w=-k2a+2a-8β-γ,          A0=-224β-ab,          A1=0,          B1=524β-ab.

Placing values in Eqs. (10) and (22), yields the following solution

Θ(χ)=24β-a(sin(χ)-2cos(χ))b(cos(χ)+2sin(χ))

and

q6(x,t)=(24β-a(sin(x+2akαtα)-2cos(x+2akαtα))b(cos(x+2akαtα)+2sin(x+2akαtα)))×ei(-kx+(wα)tα). (23)

Set 7:

One obtains p = [2 − i,−2 − i,−1,1] and q = [i,i,i,i], so Eq. (6) turns to

Ψ(χ)=-2sin(χ)-cos(χ)sin(χ). (24)

We also obtain

k=k,w=-k2a+2a-8β-γ,          A0=224β-ab,          A1=0,           B1=524β-ab.

Placing values in Eqs. (10) and (24), yields the following solution

Θ(χ)=-24β-a(sin(χ)+2cos(χ))b(-cos(χ)+2sin(χ)),

and

q7(x,t)=(-24β-a(sin(x+2akαtα)+2cos(x+2akαtα))b(-cos(x+2akαtα)+2sin(x+2akαtα)))×ei(-kx+(wα)tα). (25)

Set 8:

One obtains p = [2,0,1,−1] and q = [1,0,1,−1], so Eq. (6) turns to

Ψ(χ)=cosh(χ)+sinh(χ)sinh(χ). (26)

We also obtain

k=k,          w=-k2a-2a+8β-γ,          A0=-24β-ab,          A1=24β-ab,          B1=0.

Placing values in Eqs. (10) and (26), yields the following solution

Θ(χ)=cosh(χ)24β-absinh(χ),

and

q8(x,t)=(cosh(x+2akαtα)24β-absinh(x+2akαtα))×ei(-kx+(wα)tα). (27)

Figures 1(a-d) show numerical simulations of Eq. (27) for α = 1, 0.7, 0.5, 0.3, arbitrarily chosen.

FIGURE 1 3D Plot soliton solution related to Eq. (29). 

Set 9:

One obtains p = [−1,3,1,−1] and q = [1,−1,1,−1], so Eq. (6) turns to

Ψ(χ)=cosh(χ)-2sinh(χ)sinh(χ). (28)

We also obtain

k=k,          w=-k2a-2a+8β-γ,          A0=-224β-ab,          A1=0,          B1=-324β-ab.

Placing values in Eqs. (10) and (28), yields the following solution

Θ(χ)=-24β-a(-sinh(χ)+2cosh(χ))b(cosh(χ)-2sinh(χ))

and

q9(x,t)=(-24β-a(-sinh(x+2akαtα)+2cosh(x+2akαtα))b(cosh(x+2akαtα)-2sinh(x+2akαtα)))×ei(-kx+(wα)tα). (29)

Figures 2(a-d) show numerical simulations of Eq. (29) for α = 1, 0.7, 0.5, 0.3, arbitrarily chosen.

FIGURE 2 3D Plot soliton solution related to Eq. (29). 

Set 10:

One obtains p = [i,i,1,1] and q = [i,i,i,i], so Eq. (6) turns to

Ψ(χ)=-sin(χ)cos(χ). (30)

We also obtain

k=k,          w=-k2a-4a+16β-γ,          A0=0,          A1=-24β-ab,          B1=-24β-ab

Placing values in Eqs. (10) and (30), yields the following solution

Θ(χ)=24β-abcos(χ)sin(χ)

and

q10(x,t)=(24β-abcos(x+2akαtα)sin(x+2akαtα))×ei(-kx+(wα)tα). (31)

Figures 3(a-d) show numerical simulations of Eq. (31) for α = 1, 0.7, 0.5, 0.3, arbitrarily chosen.

FIGURE 3 3D Plot soliton solution related to Eq. (31). 

Set 11:

One obtains p = [1,1,−1,1] and q = [1,−1,1,−1], so Eq. (6) turns to

Ψ(χ)=-cosh(χ)sinh(χ). (32)

We also obtain

k=32β-8a-γ-wa,          w=w,          A0=0,          A1=-24β-ab,          B1=-24β-ab.

Placing values in Eqs. (10) and (32), yields the following solution

Θ(χ)=24β-a(coth2(χ)+1)bcoth(χ),

and

q11(x,t)=(24β-a(coth2(x+2akαtα)+1)bcoth(x+2akαtα))×ei(-kx+(wα)tα). (33)

Figures 4(a-d) show numerical simulations of Eq. (33) for α = 1, 0.7, 0.5, 0.3, arbitrarily chosen.

FIGURE 4 3D Plot soliton solution related to Eq. (33). 

4. Numerical simulations

In this work, we obtained different numerical solutions considering different alfa orders to obtain soliton solutions. The numerical solutions show that the change of fractional order modify the nature of the solution, and has a huge influence on the nonlinear propagation of the solitons. The analytical solutions allow graphing soliton solutions of type dark, bright, singular or combinations. Results showed that when the time derivative decreases, the amplitude of the solitons also decreases. It happens due to the decrease in velocity of the soliton, the order α characterizing the existence of the fractional structures on the system. The new analytical soliton solutions obtained in this paper have not been reported in the literature so far. Classical soliton solutions are recovered in the limit when α → 1.

5. Conclusion

In this work, we consider the generalized exponential rational function method to obtain approximate soliton solutions of the conformable Ginzburg-Landau equation with Kerr law nonlinearity. The Ginzburg-Landau equation describes the optical soliton propagation through a wide range of waveguides such as crystals, optical metamaterials, optical fibers and optical couplers. These soliton play an important and key role for information transfer via optical fibers.

The results showed that the generalized exponential rational function method is a promising approach to integrate the Ginzburg-Landau equation. We believe that this method also can be extrapolated to other nonlinear problems which arise in the theory of solitons.

Acknowledgments

The authors are grateful to all of the anonymous reviewers for their valuable suggestions. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Catedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

References

1. A. Biswas et al., Optik-International Journal for Light and Electron Optics 158 (2018) 399-415. [ Links ]

2. G. Akram, N. Mahak, Optik 164 (2018) 210-217. [ Links ]

3. S.T.R. Rizvi, K. Ali, M. Salman, B. Nawaz, M. Younis, OptikInternational Journal for Light and Electron Optics 149 (2017) 59-62. [ Links ]

4. A.I. Aliyu, A. Yusuf, D. Baleanu, Optik-International Journal for Light and Electron Optics 158 (2018) 368-375. [ Links ]

5. A. Biswas et al., Optik 160 (2018) 44-60. [ Links ]

6. H. Rezazadeh, Optik 167 (2018) 218-227. [ Links ]

7. S. Arshed, Optik 160 (2018) 322-332. [ Links ]

8. A.H. Arnous, A.R. Seadawy, R.T. Alqahtani, A. Biswas, Optical solitons with complex Ginzburg-Landau equation by modified simple equation method 144 (2017) 475-480. [ Links ]

9. M. Mirzazadeh et al., Nonlinear Dynamics 85 (2016) 19792016. [ Links ]

10. A.J.A.M. Jawad, M.D. Petković, A. Biswas, Applied Mathematics and Computation 217 (2010) 869-877. [ Links ]

11. K. Khan, M.A. Akbar, Ain Shams Engineering Journal 4 (2013) 903-909. [ Links ]

12. H.M. Liu, Chaos, Solitons & Fractals 23 (2005) 573-576. [ Links ]

13. R. Sassaman, A. Heidari, A. Biswas, Journal of the Franklin Institute 347 (2010) 1148-1157. [ Links ]

14. Y. Chen, Q. Wang, Chaos, Solitons & Fractals 24 (2005) 745-757. [ Links ]

15. E. Fan, J. Zhang, Physics Letters A 305 (2002) 383-392. [ Links ]

16. E. Aksoy, M. Kaplan, A. Bekir, Waves in random and complex media 26 (2016) 142-151. [ Links ]

17. B. Ghanbari, Eur. Phys. J. Plus. 133 (2018) 1-19. [ Links ]

18. A. Milovanov, J. Rasmussen, Phys. Lett. A. 337 (2005) 75-80. [ Links ]

19. M. Alquran, I. Jaradat, Nonlinear Dynamics 91 (2018) 23892395. [ Links ]

20. M.A. Abdou, A.A. Soliman, A. Biswas, M. Ekici, Q. Zhou, S.P. Moshokoa, Optik 171 (2018) 463-467. [ Links ]

21. M. Younis, H. ur Rehman, S.T.R. Rizvi, S.A. Mahmood, Superlattices and Microstructures 104 (2017) 525-531. [ Links ]

22. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, Journal of Computational and Applied Mathematics 264 (2014) 65-70. [ Links ]

23. F. Usta, An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 8 (2018) 176-182. [ Links ]

24. N. Raza, Optical and Quantum Electronics 50 (2018) 1-17. [ Links ]

25. T. Abdeljawad, Journal of Computational and Applied Mathematics 279 (2015) 57-66. [ Links ]

26. N. Benkhettou, S. Hassani, D.F. Torres, Journal of King Saud University-Science 28 (2016) 93-98. [ Links ]

27. Y. Cenesiz, D. Baleanu, A. Kurt, O. Tasbozan, Waves in random and complex media 27 (2017) 103-116. [ Links ]

28. A. Atangana, D. Baleanu, Open Mathematics 13 (2015) 1-21. [ Links ]

29. K. Hosseini, P. Mayeli, R. Ansari, Optik-International Journal for Light and Electron Optics 130 (2017) 737-742. [ Links ]

30. E. Ünal, A. Gökdogan, Journal for Light and Electron Optics 128 (2017) 264-273. [ Links ]

31. A. Biswas, Prog. Electromagn. Res. 96 (2009) 1-7. [ Links ]

Received: August 14, 2018; Accepted: September 29, 2018

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License