- Email: [email protected]
Resonant optical solitons with fractional temporal evolution by modified extended direct algebraic method
Optik - International Journal for Light and Electron Optics 181 (2019) 1075–1079
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Resonant optical solitons with fractional temporal evolution by modified extended direct algebraic method
T
Malwe Boudoue Huberta, Mibaile Justinb, Gambo Betchewea,b, Serge Y. Dokac, ⁎ T.C. Kofaned, Anjan Biswase,f,g, Salam Khane, Mehmet Ekicih, , Seithuti P. Moshokoag, i Milivoj R. Belic a
Department of Physics, Faculty of Science, The University of Maroua, P.O. Box 814, Cameroon Higher Teachers’ Training College of Maroua, The University of Maroua, P.O. Box 55, Cameroon c Department of Physics, Faculty of Science, The University of Ngaoundere, P.O. Box 454, Cameroon d Department of Physics, Faculty of Science, The University of Yaounde-I, P.O. Box 812, Cameroon e Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762-7500, USA f Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia g Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa h Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey i Institute of Physics Belgrade, Pregrevica 118, 11080 Zemun, Serbia b
ARTICLE INFO
ABSTRACT
OCIS: 060.2310 060.4510 060.5530 190.3270 190.4370
This paper employs modified extended direct algebraic method to recover bright, dark and singular solitons for resonant nonlinear Schrödinger's equation that is studied with dual-power law media. Singular periodic solutions also emerge as a byproduct of this integration scheme.
Keywords: Solitons Dual-power law Fractional temporal evolution
1. Introduction Optical solitons with fractional temporal evolution has far reaching applications in telecommunications industry. One major factor is addressing the issue of Internet bottleneck. This can be controlled by slowing down temporal evolution process, which can be achieved by the aid of fractional temporal evolution. Thus, Internet traffic can be controlled in one direction while in the other direction traffic flow remains uninterrupted. This phenomena is applicable across the globe to control the effect of Internet bottleneck. The current paper will address this issue with resonant optical solitons having dual-power law nonlinearity in presence of several perturbation terms that appear with full nonlinearity. The modified direct extended algebraic method applied to the model yields bright and singular optical soliton solutions. This is one of many mathematical algorithms that integrates such models [1–16]. The existence criteria of these solitons are also presented.
⁎
Corresponding author. E-mail address: [email protected] (M. Ekici).
https://doi.org/10.1016/j.ijleo.2018.12.181 Received 15 October 2018; Received in revised form 25 December 2018; Accepted 29 December 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 181 (2019) 1075–1079
M.B. Hubert et al.
1.1. Governing equation The resonant NLSE with perturbation terms and fractional time evolution is in the form given by [2–4]
iDtµ q + aqxx + b
|q|xx q + (c1 |q|2n + c2 |q|4n ) q = i { qx + (|q|2n q)x + (|q|2n ) x q + |q|
|q|2n qx } +
* qxx |q|2
q2 ,
(1)
where Dtµ represents the conformable derivative operator along with 0 < μ ≤ 1, a is the coefficient of group velocity dispersion (GVD), the two nonlinear forms are the coefficients of c1, c2 that combine together and balance with GVD in a delicate manner for sustaining solitons. In (1), δ accounts for inter-modal dispersion, λ represents the self-steepening coefficient, ν and θ are form nonlinearity dispersion and the coefficient σ is form Madeling fluids that makes the model resonant. 2. Mathematical preliminaries To start off with modified extended direct algebraic scheme, the hypothesis is picked as
q (x , t ) = V ( ) e i
(2)
(x , t ) ,
where the component V(ξ) represents the shape of the pulse,
+ 2(a + ) µ
=x+
t µ,
(3)
and the phase is put in the form
(x , t ) =
x+
µ
tµ +
0.
(4)
Substituting Eq. (2) into Eq. (1) implies
(a + b
)V
( +
2
+a
2) V
(5)
) V 2n + 1 + c2 V 4n+ 1 = 0,
+ (c 1
along with the constraint condition
(2n + 1) + 2nµ +
Using the transformation V = U
(a + b
(6)
= 0.
)(2nUU + (1
1/2n
, Eq. (5) leads to
2n)(U )2)
4n2 ( +
+a
2
2) U 2
+ 4n2 (c1
) U 3 + 4n2c2 U 4 = 0.
(7)
3. Modified extended direct algebraic scheme The section will adopt the modified direct extended algebraic mechanism [1,5,6,10,12–14] for deriving solitons to the governing resonant NLSE (1). For this goal, the formal solution of (7) is given as N
U( ) =
ai
i(
),
(8)
i= N
where ai are real constants to be detected such that aN ≠ 0, and N is a positive integer that will be fixed. Also, ϕ(ξ) satisfies the equation [10]
( )=
c0 + c1 ( ) + c2
2(
) + c3
3(
) + c4
4(
) + c5
5(
) + c6
6(
),
(9)
where ci are constants and can be discussed as in [15]. Employing the balance process, one acquires N = 1. Thus, the solution (8) can be rewritten as
U ( ) = a 1 ( ( ))
1
(10)
+ a 0 + a1 ( ).
Inserting (9) and (10) into (7) and utilizing [7–9,11,15], the solutions of (1) can be procured in the forms: Case-1: c0 = c1 = c3 = c5 = c6 = 0,
•a
−1 =
0, a0 = 0, a1 = a1,
=
=
c22a12 +
2 bc
4
c4 + 4 2n2c2 a12 + 2 c4 (2 n + 1)
2 bnc
4
2
4 n2c2 a12 + 2 anc4 + 2 bnc4 + ac4 + bc4 , c4 (2 n + 1) 1076
c4 n
,
Optik - International Journal for Light and Electron Optics 181 (2019) 1075–1079
M.B. Hubert et al.
For c2 > 0 and c4 < 0, bright soliton to the governing model (1) is acquired as
c2 sech c4
q1,1 (x , t ) = a1
+ 2(a + ) µ t µ
c2 x +
1/2n
ei
(x , t ) .
ei
(x , t ) ,
(11)
If c2 < 0 and c4 > 0, the recovered periodic solutions are:
q1,2 (x , t ) = a1
c2 sec c4
c2 x +
+ 2(a + ) µ t µ
q13 (x , t ) = a1
c2 csc c4
c2 x +
+ 2(a + ) µ t µ
1/2n
(12)
and
c22
Case-2: c1 = c3 = c5 = c6 = 0 and c0 =
4c4
1/2n
ei
(x , t ) .
(13)
,
• a
1
= 0,
=
1 ( 2 c12n2 4(n + 1)2c22 c2
2 2
2
2n 4c2 (1 + n)
4n3
8
2 4n2
2
c2 c12 + 4 c2 n
2 2
2 c2 n
c1
+8
1 ( 2 n3c12 + 4 n3 2(n + 1) c22
=
+ 2 n2
•
2n
a0 =
n2c12
c1
n2
2 2
+ c1 + 2 nc1
2 4n2
2
2
4 c2 n 3n3c 1
c1 + 4 n3
3c
c1
2 n3
4
4 1
n3
2 2
2
2 c2 nc12
n2
2
2 n3
4 2
2 c2
3c n2 1
+4
4 n3
(1 + 2 n)(c1
)
c2 (1 + n)
4 n3c12 c1
c1 + 8
c4 8c 2
a1 =
4 c2 2
+ 4 c2 n
+ 2 c2
,
2 4n3
2 n2
(i)
2 4n3
+ 4 c2 2 2b
2 2
c2
c1 + 4 n2c2 2 2b + 8 nc22 2b
+ 2 c2
4 n2c2 2
2 2
4
,
2 c2 n
2
2 2
+ 2 n2
+4
3c
2 1n
c1 + 2 bc22n2
8 nc2 2
n2
),
2 2
+ 2 ac2 2n2 + 4 bc22n + 4 ac2 2n + 2 bc2 2 + 2 ac22),
κ = κ, θ = θ, δ = δ .
a
1
=
=
=
=
• a
1
c4
a1 = a1,
(ii)
c4 + 4 n2 2a12c2 + 2 n 2 bc4 + c4 (1 + 2 n)
2n
2 bc 4
,
2 bnc4 + ac4 + 2 anc4 + 4 n2c2 a12 + bc4 , c4 (1 + 2 n)
• 1
a0 = 0,
2 c2 2a12
= ,
a
c2 a1 , 2c4
=
=
c1
c4 32c2
,
= .
(1 + 2 n)(c1
)
2(1 + n) c4
1 32c2 c4
(1 + 2 n)(c1 (1 + n)
)
,
a0 =
,
a0 =
2n
2n 4c2 (1 + n)
+ c1 + 2 nc1
2n
2n 4c2 (1 + n)
+ c1 + 2 nc1
,
,
(iii) κ = κ, θ = θ, δ = δ .
a1 = 0, (iv) κ = κ, θ = θ, δ = δ .
1077
Optik - International Journal for Light and Electron Optics 181 (2019) 1075–1079
M.B. Hubert et al.
By the aid of the cases (i), (ii), (iii) and (iv), one recovers the following solutions: If c2 > 0 and c4 > 0, periodic solution is found as
q2,1 (x , t ) = a
1
2c4 cot c2
c2 2
c2 tan 2c4
+ a0 + a1
1/2n
c2 2
ei
(x , t ) .
(14)
For c2 < 0 and c4 > 0, dark-singular combo soliton is revealed as
q2,2 (x , t ) = a
1
2c4 coth c2
c2 2
c2 tanh 2c4
+ a0 + a1
c2 2
1/2n
ei
(x , t ) .
(15)
Case-3: c3 = c4 = c5 = c6 = 0,
• a
1
=
c0 (2 nc1 c1 c2 (n + 1)
2n
+ c1
2n
),
a 0 = 0,
a1 = 0,
κ = κ, θ = θ, δ = δ .
The solution of this case can be expressed as
q3 (x , t ) = a
4c2 c0 sinh[ c2 ] 2c2
1
1/2n
c1 2c2
ei
(x , t ) .
(16)
Case-4: c0 = c1 = c2 = c5 = c6 = 0,
• a
1
= 0,
a 0 = 0,
a1 =
c4 (2 nc1
2n c3 c2 (n + 1)
2n
+ c1 )
,
κ = κ, θ = θ, δ = δ .
The solution in this case is
q4 (x , t ) =
4a1 c3 4c4
1/2n
ei
c32 2
(x , t ) ,
(17)
as long as c4 > 0. Case-5: c0 = c1 = c5 = c6 = 0,
• a
1
= 0,
a 0 = 0,
a1 =
c4 (2 nc1
2n + c1 c3 c2 (n + 1)
2n
)
,
κ = κ, θ = θ, δ = δ .
The solutions of this case are [11]: When c32 = 4c2 c4 and 2α2 < 9β, dark and singular optical soliton solutions are procured respectively
a1
c2 c 1 + tanh( 2 ) c3 2
1/2n
q5,11 (x , t ) =
a1
c2 c 1 + coth( 2 ) c3 2
1/2n
q5,12 (x , t ) = For
c32
If
(x , t ) ,
ei
(x , t ) .
(18) (19)
> 4c2 c4 and c2 > 0, bright soliton is derived as
q5,2 (x , t ) = a1 c32
ei
2c2sech( c2 (x
vt))
c3sech( c2 )
1/2n
ei
(x , t ) .
(20)
> 4c2 c4 and c2 < 0, the model has the solutions given by
q5,31 (x , t ) = a1
2c2sec( c3sec(
c2 ) c2 )
1/2n
ei
(x , t ) ,
(21) 1078
Optik - International Journal for Light and Electron Optics 181 (2019) 1075–1079
M.B. Hubert et al.
q5,32 (x , t ) = a1 Whenever
c32
2c2 csc(
1/2n
c2 )
c3 csc(
ei
c2 )
(x , t ) .
(22)
< 4c2 c4 and c2 > 0, it is a singular optical soliton
q5,4 (x , t ) = a1
2c2 csch( c2 ) c3 csch( c2 )
ei
(x , t ) .
(23)
4. Conclusions This paper recovered dark and singular resonant optical solitons with dual-power law media having fractional temporal evolution. The modified extended direct algebraic scheme was applied to obtain such solitons. The results will be very useful in telecommunication industry in order to control Internet bottleneck effect. The future prospects hold very strong in this area of research. Later, the model will be considered with DWDM topology that will enable parallel communication of soliton molecules without the effect of such bottleneck. These studies are under way and the results will be reported very soon. Moreover, the same technology will also be applied to optical couplers and metamaterials that will yield fruitful results for our modern day daily communications. Conflict of interest The authors also declare that there is no conflict of interest. Acknowledgements The ninth author (SPM) would like to thank the research support provided by the Department of Mathematics and Statistics at Tshwane University of Technology and the support from the South African National Foundation under Grant Number 92052 IRF1202210126. The research work of tenth author (MRB) was supported by the grant NPRP 8-028-1-001 from QNRF and he is thankful for it. References [1] M. Arshad, A.R. Seadawy, D. Lu, J. Wang, Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations, Results Phys. 6 (2016) 1136–1145. [2] A. Biswas, M.O. Al-Amr, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, S.P. Moshokoa, M. Belic, Resonant optical solitons with dual power-law nonlinearity and fractional temporal evolution, Optik 165 (2018) 233–239. [3] A. Biswas, Q. Zhou, S.P. Moshokoa, H. Triki, M. Belic, R.T. Alqahtani, Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations, Optik 145 (2017) 14–17. [4] A. Biswas, M. Mirzazadeh, H. Triki, Q. Zhou, M.Z. Ullah, S.P. Moshokoa, M. Belic, Perturbed resonant 1-soliton solution with anti-cubic nonlinearity by Riccati–Bernoulli sub-ODE method, Optik 156 (2018) 346–350. [5] M.B. Hubert, M. Justin, G. Betchewe, S.Y. Doka, A. Biswas, Q. Zhou, M. Ekici, S.P. Moshokoa, M. Belic, Optical solitons with modified extended direct algebraic method for quadratic–cubic nonlinearity, Optik 162 (2018) 161–171. [6] D. Lu, A.R. Seadawy, M. Arshad, J. Wang, New solitary wave solutions of (3+1)-dimensional nonlinear extended Zakharov–Kuznetsov and modified KdV–Zakharov–Kuznetsov equations and their applications, Results Phys. 7 (2017) 899–909. [7] J. Nickel, H.W. Schurmann, V.S. Serov, Some elliptic travelling wave solution to the Novikkov–Veselov equation, Proc. Prog. Electromagn. Res. Sympos. 61 (2006) 323–354. [8] J. Nickel, Elliptic solutions to a generalized BBM equation, Phys. Lett. A 364 (3-4) (2007) 221–226. [9] H.W. Schurmann, Travelling-wave solutions of the cubic–quintic nonlinear Schrödinger equation, Phys. Rev. E 54 (4) (1996) 4312–4332. [10] A.R. Seadawy, M. Arshad, D. Lu, Stability analysis of new exact traveling wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems, Eur. Phys. J. Plus 132 (2017) 162. [11] Sirendaoreji, Auxiliary method and new solutions of Klein–Gordon equations, Chaos Solitons Fract. 31 (4) (2007) 943–950. [12] A.A. Soliman, The modified extended direct algebraic method for solving nonlinear partial differential equations, Int. J. Nonlinear Sci. 6 (2) (2008) 136–144. [13] N. Taghizadeh, M.N. Foumani, Using a reliable method for higher dimensional of the fractional Schrödinger equation, J. Math. 48 (1) (2016) 11–18. [14] M. Younis, H. Rehman, M. Iftikhar, Computational examples of a class of fractional order nonlinear evolution equations using modified extended direct algebraic method, J. Comput. Methods Sci. Eng. 15 (3) (2015) 359–365. [15] L.H. Zhang, Travelling wave solutions for the generalized Zakharov–Kuznetsov equation with higher-order nonlinear terms, Appl. Math. Comput. 208 (1) (2009) 144–155. [16] Q. Zhou, L. Liu, H. Zhang, M. Mirzazadeh, A.H. Bhrawy, E. Zerrad, S. Moshokoa, Anjan Biswas, Dark and singular optical solitons with competing nonlocal nonlinearities, Opt. Appl. 46 (1) (2016) 79–86.
1079
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Resonant optical solitons with fractional temporal evolution by modified extended direct algebraic method
T
Malwe Boudoue Huberta, Mibaile Justinb, Gambo Betchewea,b, Serge Y. Dokac, ⁎ T.C. Kofaned, Anjan Biswase,f,g, Salam Khane, Mehmet Ekicih, , Seithuti P. Moshokoag, i Milivoj R. Belic a
Department of Physics, Faculty of Science, The University of Maroua, P.O. Box 814, Cameroon Higher Teachers’ Training College of Maroua, The University of Maroua, P.O. Box 55, Cameroon c Department of Physics, Faculty of Science, The University of Ngaoundere, P.O. Box 454, Cameroon d Department of Physics, Faculty of Science, The University of Yaounde-I, P.O. Box 812, Cameroon e Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762-7500, USA f Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia g Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa h Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey i Institute of Physics Belgrade, Pregrevica 118, 11080 Zemun, Serbia b
ARTICLE INFO
ABSTRACT
OCIS: 060.2310 060.4510 060.5530 190.3270 190.4370
This paper employs modified extended direct algebraic method to recover bright, dark and singular solitons for resonant nonlinear Schrödinger's equation that is studied with dual-power law media. Singular periodic solutions also emerge as a byproduct of this integration scheme.
Keywords: Solitons Dual-power law Fractional temporal evolution
1. Introduction Optical solitons with fractional temporal evolution has far reaching applications in telecommunications industry. One major factor is addressing the issue of Internet bottleneck. This can be controlled by slowing down temporal evolution process, which can be achieved by the aid of fractional temporal evolution. Thus, Internet traffic can be controlled in one direction while in the other direction traffic flow remains uninterrupted. This phenomena is applicable across the globe to control the effect of Internet bottleneck. The current paper will address this issue with resonant optical solitons having dual-power law nonlinearity in presence of several perturbation terms that appear with full nonlinearity. The modified direct extended algebraic method applied to the model yields bright and singular optical soliton solutions. This is one of many mathematical algorithms that integrates such models [1–16]. The existence criteria of these solitons are also presented.
⁎
Corresponding author. E-mail address: [email protected] (M. Ekici).
https://doi.org/10.1016/j.ijleo.2018.12.181 Received 15 October 2018; Received in revised form 25 December 2018; Accepted 29 December 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 181 (2019) 1075–1079
M.B. Hubert et al.
1.1. Governing equation The resonant NLSE with perturbation terms and fractional time evolution is in the form given by [2–4]
iDtµ q + aqxx + b
|q|xx q + (c1 |q|2n + c2 |q|4n ) q = i { qx + (|q|2n q)x + (|q|2n ) x q + |q|
|q|2n qx } +
* qxx |q|2
q2 ,
(1)
where Dtµ represents the conformable derivative operator along with 0 < μ ≤ 1, a is the coefficient of group velocity dispersion (GVD), the two nonlinear forms are the coefficients of c1, c2 that combine together and balance with GVD in a delicate manner for sustaining solitons. In (1), δ accounts for inter-modal dispersion, λ represents the self-steepening coefficient, ν and θ are form nonlinearity dispersion and the coefficient σ is form Madeling fluids that makes the model resonant. 2. Mathematical preliminaries To start off with modified extended direct algebraic scheme, the hypothesis is picked as
q (x , t ) = V ( ) e i
(2)
(x , t ) ,
where the component V(ξ) represents the shape of the pulse,
+ 2(a + ) µ
=x+
t µ,
(3)
and the phase is put in the form
(x , t ) =
x+
µ
tµ +
0.
(4)
Substituting Eq. (2) into Eq. (1) implies
(a + b
)V
( +
2
+a
2) V
(5)
) V 2n + 1 + c2 V 4n+ 1 = 0,
+ (c 1
along with the constraint condition
(2n + 1) + 2nµ +
Using the transformation V = U
(a + b
(6)
= 0.
)(2nUU + (1
1/2n
, Eq. (5) leads to
2n)(U )2)
4n2 ( +
+a
2
2) U 2
+ 4n2 (c1
) U 3 + 4n2c2 U 4 = 0.
(7)
3. Modified extended direct algebraic scheme The section will adopt the modified direct extended algebraic mechanism [1,5,6,10,12–14] for deriving solitons to the governing resonant NLSE (1). For this goal, the formal solution of (7) is given as N
U( ) =
ai
i(
),
(8)
i= N
where ai are real constants to be detected such that aN ≠ 0, and N is a positive integer that will be fixed. Also, ϕ(ξ) satisfies the equation [10]
( )=
c0 + c1 ( ) + c2
2(
) + c3
3(
) + c4
4(
) + c5
5(
) + c6
6(
),
(9)
where ci are constants and can be discussed as in [15]. Employing the balance process, one acquires N = 1. Thus, the solution (8) can be rewritten as
U ( ) = a 1 ( ( ))
1
(10)
+ a 0 + a1 ( ).
Inserting (9) and (10) into (7) and utilizing [7–9,11,15], the solutions of (1) can be procured in the forms: Case-1: c0 = c1 = c3 = c5 = c6 = 0,
•a
−1 =
0, a0 = 0, a1 = a1,
=
=
c22a12 +
2 bc
4
c4 + 4 2n2c2 a12 + 2 c4 (2 n + 1)
2 bnc
4
2
4 n2c2 a12 + 2 anc4 + 2 bnc4 + ac4 + bc4 , c4 (2 n + 1) 1076
c4 n
,
Optik - International Journal for Light and Electron Optics 181 (2019) 1075–1079
M.B. Hubert et al.
For c2 > 0 and c4 < 0, bright soliton to the governing model (1) is acquired as
c2 sech c4
q1,1 (x , t ) = a1
+ 2(a + ) µ t µ
c2 x +
1/2n
ei
(x , t ) .
ei
(x , t ) ,
(11)
If c2 < 0 and c4 > 0, the recovered periodic solutions are:
q1,2 (x , t ) = a1
c2 sec c4
c2 x +
+ 2(a + ) µ t µ
q13 (x , t ) = a1
c2 csc c4
c2 x +
+ 2(a + ) µ t µ
1/2n
(12)
and
c22
Case-2: c1 = c3 = c5 = c6 = 0 and c0 =
4c4
1/2n
ei
(x , t ) .
(13)
,
• a
1
= 0,
=
1 ( 2 c12n2 4(n + 1)2c22 c2
2 2
2
2n 4c2 (1 + n)
4n3
8
2 4n2
2
c2 c12 + 4 c2 n
2 2
2 c2 n
c1
+8
1 ( 2 n3c12 + 4 n3 2(n + 1) c22
=
+ 2 n2
•
2n
a0 =
n2c12
c1
n2
2 2
+ c1 + 2 nc1
2 4n2
2
2
4 c2 n 3n3c 1
c1 + 4 n3
3c
c1
2 n3
4
4 1
n3
2 2
2
2 c2 nc12
n2
2
2 n3
4 2
2 c2
3c n2 1
+4
4 n3
(1 + 2 n)(c1
)
c2 (1 + n)
4 n3c12 c1
c1 + 8
c4 8c 2
a1 =
4 c2 2
+ 4 c2 n
+ 2 c2
,
2 4n3
2 n2
(i)
2 4n3
+ 4 c2 2 2b
2 2
c2
c1 + 4 n2c2 2 2b + 8 nc22 2b
+ 2 c2
4 n2c2 2
2 2
4
,
2 c2 n
2
2 2
+ 2 n2
+4
3c
2 1n
c1 + 2 bc22n2
8 nc2 2
n2
),
2 2
+ 2 ac2 2n2 + 4 bc22n + 4 ac2 2n + 2 bc2 2 + 2 ac22),
κ = κ, θ = θ, δ = δ .
a
1
=
=
=
=
• a
1
c4
a1 = a1,
(ii)
c4 + 4 n2 2a12c2 + 2 n 2 bc4 + c4 (1 + 2 n)
2n
2 bc 4
,
2 bnc4 + ac4 + 2 anc4 + 4 n2c2 a12 + bc4 , c4 (1 + 2 n)
• 1
a0 = 0,
2 c2 2a12
= ,
a
c2 a1 , 2c4
=
=
c1
c4 32c2
,
= .
(1 + 2 n)(c1
)
2(1 + n) c4
1 32c2 c4
(1 + 2 n)(c1 (1 + n)
)
,
a0 =
,
a0 =
2n
2n 4c2 (1 + n)
+ c1 + 2 nc1
2n
2n 4c2 (1 + n)
+ c1 + 2 nc1
,
,
(iii) κ = κ, θ = θ, δ = δ .
a1 = 0, (iv) κ = κ, θ = θ, δ = δ .
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M.B. Hubert et al.
By the aid of the cases (i), (ii), (iii) and (iv), one recovers the following solutions: If c2 > 0 and c4 > 0, periodic solution is found as
q2,1 (x , t ) = a
1
2c4 cot c2
c2 2
c2 tan 2c4
+ a0 + a1
1/2n
c2 2
ei
(x , t ) .
(14)
For c2 < 0 and c4 > 0, dark-singular combo soliton is revealed as
q2,2 (x , t ) = a
1
2c4 coth c2
c2 2
c2 tanh 2c4
+ a0 + a1
c2 2
1/2n
ei
(x , t ) .
(15)
Case-3: c3 = c4 = c5 = c6 = 0,
• a
1
=
c0 (2 nc1 c1 c2 (n + 1)
2n
+ c1
2n
),
a 0 = 0,
a1 = 0,
κ = κ, θ = θ, δ = δ .
The solution of this case can be expressed as
q3 (x , t ) = a
4c2 c0 sinh[ c2 ] 2c2
1
1/2n
c1 2c2
ei
(x , t ) .
(16)
Case-4: c0 = c1 = c2 = c5 = c6 = 0,
• a
1
= 0,
a 0 = 0,
a1 =
c4 (2 nc1
2n c3 c2 (n + 1)
2n
+ c1 )
,
κ = κ, θ = θ, δ = δ .
The solution in this case is
q4 (x , t ) =
4a1 c3 4c4
1/2n
ei
c32 2
(x , t ) ,
(17)
as long as c4 > 0. Case-5: c0 = c1 = c5 = c6 = 0,
• a
1
= 0,
a 0 = 0,
a1 =
c4 (2 nc1
2n + c1 c3 c2 (n + 1)
2n
)
,
κ = κ, θ = θ, δ = δ .
The solutions of this case are [11]: When c32 = 4c2 c4 and 2α2 < 9β, dark and singular optical soliton solutions are procured respectively
a1
c2 c 1 + tanh( 2 ) c3 2
1/2n
q5,11 (x , t ) =
a1
c2 c 1 + coth( 2 ) c3 2
1/2n
q5,12 (x , t ) = For
c32
If
(x , t ) ,
ei
(x , t ) .
(18) (19)
> 4c2 c4 and c2 > 0, bright soliton is derived as
q5,2 (x , t ) = a1 c32
ei
2c2sech( c2 (x
vt))
c3sech( c2 )
1/2n
ei
(x , t ) .
(20)
> 4c2 c4 and c2 < 0, the model has the solutions given by
q5,31 (x , t ) = a1
2c2sec( c3sec(
c2 ) c2 )
1/2n
ei
(x , t ) ,
(21) 1078
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M.B. Hubert et al.
q5,32 (x , t ) = a1 Whenever
c32
2c2 csc(
1/2n
c2 )
c3 csc(
ei
c2 )
(x , t ) .
(22)
< 4c2 c4 and c2 > 0, it is a singular optical soliton
q5,4 (x , t ) = a1
2c2 csch( c2 ) c3 csch( c2 )
ei
(x , t ) .
(23)
4. Conclusions This paper recovered dark and singular resonant optical solitons with dual-power law media having fractional temporal evolution. The modified extended direct algebraic scheme was applied to obtain such solitons. The results will be very useful in telecommunication industry in order to control Internet bottleneck effect. The future prospects hold very strong in this area of research. Later, the model will be considered with DWDM topology that will enable parallel communication of soliton molecules without the effect of such bottleneck. These studies are under way and the results will be reported very soon. Moreover, the same technology will also be applied to optical couplers and metamaterials that will yield fruitful results for our modern day daily communications. Conflict of interest The authors also declare that there is no conflict of interest. Acknowledgements The ninth author (SPM) would like to thank the research support provided by the Department of Mathematics and Statistics at Tshwane University of Technology and the support from the South African National Foundation under Grant Number 92052 IRF1202210126. The research work of tenth author (MRB) was supported by the grant NPRP 8-028-1-001 from QNRF and he is thankful for it. References [1] M. Arshad, A.R. Seadawy, D. Lu, J. Wang, Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations, Results Phys. 6 (2016) 1136–1145. [2] A. Biswas, M.O. Al-Amr, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, S.P. Moshokoa, M. Belic, Resonant optical solitons with dual power-law nonlinearity and fractional temporal evolution, Optik 165 (2018) 233–239. [3] A. Biswas, Q. Zhou, S.P. Moshokoa, H. Triki, M. Belic, R.T. Alqahtani, Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations, Optik 145 (2017) 14–17. [4] A. Biswas, M. Mirzazadeh, H. Triki, Q. Zhou, M.Z. Ullah, S.P. Moshokoa, M. Belic, Perturbed resonant 1-soliton solution with anti-cubic nonlinearity by Riccati–Bernoulli sub-ODE method, Optik 156 (2018) 346–350. [5] M.B. Hubert, M. Justin, G. Betchewe, S.Y. Doka, A. Biswas, Q. Zhou, M. Ekici, S.P. Moshokoa, M. Belic, Optical solitons with modified extended direct algebraic method for quadratic–cubic nonlinearity, Optik 162 (2018) 161–171. [6] D. Lu, A.R. Seadawy, M. Arshad, J. Wang, New solitary wave solutions of (3+1)-dimensional nonlinear extended Zakharov–Kuznetsov and modified KdV–Zakharov–Kuznetsov equations and their applications, Results Phys. 7 (2017) 899–909. [7] J. Nickel, H.W. Schurmann, V.S. Serov, Some elliptic travelling wave solution to the Novikkov–Veselov equation, Proc. Prog. Electromagn. Res. Sympos. 61 (2006) 323–354. [8] J. Nickel, Elliptic solutions to a generalized BBM equation, Phys. Lett. A 364 (3-4) (2007) 221–226. [9] H.W. Schurmann, Travelling-wave solutions of the cubic–quintic nonlinear Schrödinger equation, Phys. Rev. E 54 (4) (1996) 4312–4332. [10] A.R. Seadawy, M. Arshad, D. Lu, Stability analysis of new exact traveling wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems, Eur. Phys. J. Plus 132 (2017) 162. [11] Sirendaoreji, Auxiliary method and new solutions of Klein–Gordon equations, Chaos Solitons Fract. 31 (4) (2007) 943–950. [12] A.A. Soliman, The modified extended direct algebraic method for solving nonlinear partial differential equations, Int. J. Nonlinear Sci. 6 (2) (2008) 136–144. [13] N. Taghizadeh, M.N. Foumani, Using a reliable method for higher dimensional of the fractional Schrödinger equation, J. Math. 48 (1) (2016) 11–18. [14] M. Younis, H. Rehman, M. Iftikhar, Computational examples of a class of fractional order nonlinear evolution equations using modified extended direct algebraic method, J. Comput. Methods Sci. Eng. 15 (3) (2015) 359–365. [15] L.H. Zhang, Travelling wave solutions for the generalized Zakharov–Kuznetsov equation with higher-order nonlinear terms, Appl. Math. Comput. 208 (1) (2009) 144–155. [16] Q. Zhou, L. Liu, H. Zhang, M. Mirzazadeh, A.H. Bhrawy, E. Zerrad, S. Moshokoa, Anjan Biswas, Dark and singular optical solitons with competing nonlocal nonlinearities, Opt. Appl. 46 (1) (2016) 79–86.
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