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Highly dispersive optical solitons with non-local nonlinearity by exp-function
Optik - International Journal for Light and Electron Optics 186 (2019) 288–292
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Highly dispersive optical solitons with non-local nonlinearity by exp-function
T
⁎
Anjan Biswasa,b,c, Mehmet Ekicid, , Abdullah Sonmezoglud, Milivoj R. Belice a
Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762-7500, USA Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa d Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey e Institute of Physics Belgrade, Pregrevica 118, 11080 Zemun, Serbia b c
A R T IC LE I N F O
ABS TRA CT
OCIS: 060.2310 060.4510 060.5530 190.3270 190.4370Keywords: Dispersive solitons Non-local nonlinearity exp−function
This paper recovers singular and bright-singular highly dispersive optical solitons, having nonlocal form of nonlinearity, by the application of exp-function integration scheme. The existence criteria for these solitons are also listed in the work.
1. Introduction The advancement of optical solitons is primarily due to deep mathematical theories that are applicable to study this dynamics. One of the aspects of soliton dynamics is highly dispersive optical solitons which appears when inter-modal dispersion (IMD), thirdorder dispersion (3OD), fourth-order dispersion (4OD), fifth-order dispersion (5OD) and sixth-order dispersion (6OD) terms are taken into account in addition to the presence of the usual group-velocity dispersion (GVD) [1–10]. Several mathematical techniques have been applied to handle such highly dispersive optical solitons. These are the method of undetermined coefficients, F-expansion scheme, extended Jacobi's elliptic function expansion and lately the exp-function method. While the first method turned out to be an epic failure to retrieve soliton solutions [5], the second and third methods have proved to be a grand success in retrieving soliton solutions to the governing nonlinear Schrödinger's equation (NLSE) with six dispersion terms [1–4,6–8]. Finally, the exp-function scheme has proved to be successful with quadratic–cubic law and cubic–quintic–septic law of nonlinearity, thus far [9,10]. It has also been successfully applied to other physical systems such as birefringent fibers, water waves, oblique solitons and other such [11–15]. This paper therefore encouragingly studies highly dispersive optical solitons with non-local law of nonlinearity using exp-function. The striking details are presented in the rest of the paper after an intro to the governing model. 1.1. Governing model The dimensionless form of NLSE with non-local nonlinearity in presence of dispersion terms of all orders is [1–10]:
⁎
Corresponding author. E-mail address: [email protected] (M. Ekici).
https://doi.org/10.1016/j.ijleo.2019.04.082 Received 18 February 2019; Accepted 16 April 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 186 (2019) 288–292
A. Biswas, et al.
iq t + ia1 qx + a2 qxx + ia3qxxx + a4 qxxxx + ia5qxxxxx + a6 qxxxxxx + b (|q|2 ) xx q = 0.
(1)
Here, in (1), q(x, t) represents soliton molecules where x and t are independent spatial and temporal variables respectively. The first term represents linear temporal evolution, while a1, a2, a3, a4, a5 and a6 are coefficients of IMD, GVD, 3OD, 4OD, 5OD and 6OD respectively. Finally, b is the coefficient of non–local nonlinearity. The complex–valued function q(x, t) is the wave profile that represents the dependent variable. The coefficients are all real–valued constants and i = −1 . 2. Preliminaries To start off, the starting point is the hypothesis given by
q (x , t ) = g (ς ) eiϕ (x , t )
(2)
ς = x − vt
(3)
where
with v denotes the velocity of the soliton. Next, the phase component is defined as
ϕ (x , t ) = −κx + ωt + θ
(4)
where κ is the soliton frequency, ω is its wave number and θ is the phase constant. Once putting (2) into (1) real and imaginary parts yield
− (ω + κ (κ (a2 + κ (a3 + a6 κ 3 − κ (a4 + a5 κ ))) − a1 )) g + 2b g(g ′)2 + (a2 + κ (3a3 + κ (5κ (3a6 κ − 2a5) − 6a4 ))) g ″ + 2bg 2g ″ + (a4 + 5κ (a5 − 3a6 κ )) g (4) + a6 g (6) = 0
(5)
(v − a1 + 2a2 κ + 3a3 κ 2 − 4a4 κ 3 − 5a5 κ 4 + 6a6 κ 5) g + (−a3 + 2κ (2a4 + 5κ (a5 − 2a6 κ ))) g ″ − (a5 − 6a6 κ ) g (4) = 0
(6)
and
2
2
respectively. The notations g′ = dg/dς, g″ = d g/dς and so on are introduced. From (6), the constraint conditions are (7)
a5 = 6a6 κ a3 =
4κ (3a4 + 5a5 κ ) 3
(8)
and thus the soliton speed comes up as below:
v = a1 − 2κ (a2 + 4a4 κ 2 + 8a5 κ 3).
(9)
In view of (7) and (8), the real part can be rewritten as
− κ (6ω − 6a1 κ + 6a2 κ 2 + 18a4 κ 4 + 35a5 κ 5) g + 12bκg (g ′)2 + 3κ (2a2 + κ 2 (12a4 + 25a5 κ )) g ″ + 12bκg 2g ″ + 3κ (2a4 + 5a5 κ ) g (4) + a5 g (6) = 0.
(10)
3. exp(−ψ (ς ))−expansion scheme In order to investigate highly dispersive optical solitons to the NLSE with non-local nonlinearity by exp(−ψ (ς )) -expansion [11–15], the considered hypothesis is: N
g (ς ) =
∑ γj (exp[−ψ (ς )]) j .
(11)
j=1
Here, in (11), the coefficients γj are constants to be fixed later, such that γN ≠ 0, and the function ψ(ς) holds
ψ′ (ς ) = exp[−ψ (ς )] + χ exp[ψ (ς )] + ϑ
(12)
where the solutions of (12) are: If χ ≠ 0 and ϑ2 − 4χ > 0, 2
ϑ − 4χ ⎡ ϑ2 − 4χ tanh ⎛ 2 (ς + c ) ⎞ + ϑ ⎤ ⎢ ⎥ ⎝ ⎠ ψ (ς ) = ln ⎢− ⎥. 2χ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
(13)
For χ ≠ 0 and ϑ − 4χ < 0, 2
289
Optik - International Journal for Light and Electron Optics 186 (2019) 288–292
A. Biswas, et al.
2
⎡ 4χ − ϑ2 tan ⎛ 4χ − ϑ (ς + c ) ⎞ − ϑ ⎤ 2 ⎥ ⎢ ⎝ ⎠ ψ (ς ) = ln ⎢ ⎥. 2 χ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣
(14)
When χ = 0, ϑ ≠ 0 and ϑ2 − 4χ > 0,
ϑ ⎤. ψ (ς ) = −ln ⎡ ⎢ ⎣ exp(ϑ(ς + c )) − 1 ⎥ ⎦
(15)
Whenever χ ≠ 0, ϑ ≠ 0 and ϑ − 4χ = 0, 2
ψ (ς ) = ln ⎡− ⎢ ⎣
2(ϑ(ς + c ) + 2) ⎤ . ⎥ ϑ2 (ς + c ) ⎦
(16)
However, if χ = 0, ϑ = 0 and ϑ − 4χ = 0, 2
ψ (ς ) = ln[ς + c ].
(17) 2
It is important to note that c is the integration constant. Balancing g g″ with g (10) is of the form
(6)
in (10) gives N = 2. Therefore, the solution of Eq.
g (ς ) = γ0 + γ1 exp[−ψ (ς )] + γ2 exp[−2ψ (ς )].
(18)
Inserting (18) into (10), combining up the coefficients of exp(−ψ (ς )) , and coping with the resulting system, one gets the following sets: Set-1:
γ0 = γ2 χ , γ1 = γ2 ϑ, γ2 = γ2, 30a1 κ 2 − a5 (25κ 6 + 84κ 4 (ϑ2 − 4χ ) + 35κ 2 (ϑ2 − 4χ )2 + 12(4χ − ϑ2)3) , 30κ 4 2 2 2 2 a (75κ + 168κ (ϑ − 4χ ) + 35(ϑ − 4χ ) ) a2 = 5 , 30κ a (75κ 2 + 28(ϑ2 − 4χ )) , a4 = − 5 30κ 42a5 b=− 2 , γ2 κ ω=
(19)
Set-2:
γ2 (3χ + ϑ2) , 7 γ1 = γ2 ϑ, γ2 = γ2,
γ0 =
κ (42a1 − a5 (35κ 4 + 84κ 2 (4χ − ϑ2) + 37(ϑ2 − 4χ )2)) , 42 a (105κ 4 + 168κ 2 (4χ − ϑ2) + 37(ϑ2 − 4χ )2) a2 = 5 , 42κ 2 2 a (15κ + 16χ − 4ϑ ) , a4 = − 5 6κ 42a b = − 2 5. γ2 κ ω=
(20)
By employing the first set of parameters, one can secure bright-singular combo solitons solution
γ2 χ (4χ − ϑ2)
q (x , t )= ⎛⎜ϑ cosh ⎡ ⎣ ⎝
ϑ2 − 4χ (c + ς ) 2
⎤+ ⎦
ϑ2 − 4χ sinh ⎡ ⎣
ϑ2 − 4χ (c + ς ) 2
2
⎤ ⎟⎞ ⎦⎠
2 6 4 2 2 2 2 2 3 ⎧ ⎛ 30a1 κ − a5 (25κ + 84κ (ϑ − 4χ ) + 35κ (ϑ − 4χ ) + 12(4χ − ϑ ) ) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 30 κ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
trigonometric function solution 290
(21)
Optik - International Journal for Light and Electron Optics 186 (2019) 288–292
A. Biswas, et al.
γ2 χ (4χ − ϑ2)
q (x , t )= ⎛⎜ϑ cos ⎡ ⎣ ⎝
4χ − ϑ2 (c + ς ) 2
⎤− ⎦
4χ − ϑ2 sin ⎡ ⎣
4χ − ϑ2 (c + ς ) 2
2
⎤ ⎟⎞ ⎦⎠
2 6 4 2 2 2 2 2 3 ⎧ ⎛ 30a1 κ − a5 (25κ + 84κ (ϑ − 4χ ) + 35κ (ϑ − 4χ ) + 12(4χ − ϑ ) ) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 30 κ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(22)
singular soliton solution
q (x , t ) =
2 6 4 2 2 4 6 γ2 ϑ2 ϑ(c + ς ) ⎤ ⎧ ⎛ 30a1 κ − a5 (25κ + 84κ ϑ + 35κ ϑ − 12ϑ ) ⎞ t + θ ⎫ ⎤ exp ⎡ csch2 ⎡ ⎢i ⎨−κx + ⎥ ⎬ 2 30 4 κ ⎣ ⎦ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(23)
and finally plane wave solutions
q (x , t ) =
2 6 γ2 ⎛ ϑ3 (c + ς )[ϑ(c + ς ) + 4] ⎞ ⎡i ⎧−κx + ⎛ 30a1 κ − 25a5 κ ⎞ t + θ ⎫ ⎤ ⎜4χ − ⎟ exp 2 ⎢ ⎬⎥ [ϑ(c + ς ) + 2] 30κ 4⎝ ⎝ ⎠ ⎠ ⎩ ⎭ ⎦ ⎣ ⎨ ⎜
⎟
(24)
2 6 ⎧ ⎛ 30a1 κ − 25a5 κ ⎞ t + θ ⎫ ⎤. q (x , t ) = γ2 (c + ς )2exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 30 κ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(25)
Next, by the use of the second set of parameters, one can explore singular soliton solution
⎡ γ2 ⎢ 2 ⎢ϑ + q (x , t )= 7⎢ ⎛⎜ϑ + ⎢ ⎝ ⎣
⎛ ⎜ +χ 3− 2 ⎜ ϑ2 − 4χ (c + ς ) ⎞ 2 ⎡ ⎤ ϑ+ ϑ − 4χ tanh ⎟ ⎜ 2 ⎝ ⎣ ⎦⎠ 28χ 2
⎞⎤ ⎥ ⎟⎥ ϑ2 − 4χ (c + ς ) ⎟ ⎥ ⎤⎟ ϑ2 − 4χ tanh ⎡ ⎥ 2 ⎣ ⎦⎠⎦ 14ϑ
4 2 2 2 2 ⎧ ⎛ κ (42a1 − a5 (35κ + 84κ (4χ − ϑ ) + 37(ϑ − 4χ ) )) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(26)
periodic solution
⎡ γ2 ⎢ 2 ⎢ϑ + q (x , t )= 7⎢ ⎛⎜ϑ − ⎢ ⎝ ⎣
⎛ ⎜ +χ 3− 2 ⎜ 4χ − ϑ2 (c + ς ) ⎞ 2 ⎤⎟ ϑ− 4χ − ϑ tan ⎡ ⎜ 2 ⎝ ⎣ ⎦⎠ 28χ 2
⎞⎤ ⎥ ⎟⎥ 2 ⎟ 4χ − ϑ (c + ς ) ⎤⎟⎥ 4χ − ϑ2 tan ⎡ ⎥ 2 ⎣ ⎦⎠⎦ 14ϑ
4 2 2 2 2 ⎧ ⎛ κ (42a1 − a5 (35κ + 84κ (4χ − ϑ ) + 37(ϑ − 4χ ) )) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(27)
bright-singular combo solitons solution
1 1 1 ⎞ + ⎟ q (x , t )= γ2 ϑ2 ⎛⎜ + cosh[ϑ(c + ς )] + sinh[ϑ(c + ς )] − 1 (cosh[ϑ(c + ς )] + sinh[ϑ(c + ς )] − 1)2 ⎠ ⎝7 4 2 2 4 ⎧ ⎛ κ (42a1 − a5 (35κ − 84κ ϑ + 37ϑ )) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎢i ⎨−κx + ⎥ ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(28)
and finally plane waves
q (x , t ) =
γ2 (12χ [ϑ(c + ς ) + 2]2 + ϑ2 (16 − 3ϑ(c + ς )[ϑ(c + ς ) + 4])) κ (42a1 − 35a5 κ 4 ) ⎞ exp ⎡ i ⎧−κx + ⎛ t + θ⎫ ⎤ 2 ⎢ ⎬⎥ ⎨ 28[ϑ(c + ς ) + 2] 42 ⎝ ⎠ ⎩ ⎭ ⎦ ⎣ ⎜
4 ⎧ ⎛ κ (42a1 − 35a5 κ ) ⎞ t + θ ⎫ ⎤. q (x , t ) = γ2 (c + ς )2exp ⎡ ⎢i ⎨−κx + ⎥ ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(29)
⎟
(30)
4. Conclusions This paper recovered highly dispersive optical solitons having non-local nonlinearity by exp-function. It is only singular and bright-singular combo solitons that emerged from this scheme. Surely, this algorithm fails to retrieve the necessary and important bright and dark soliton solutions to the model. Therefore, in order to get a complete spectrum of soliton solutions to the model, one must resort to additional integration algorithms such as Lie symmetry analysis, Kudryashov's scheme, G′/G-expansion scheme and several others. It is expected that these mechanisms will achieve our goal. Later on, the conservation laws will also be derived for the 291
Optik - International Journal for Light and Electron Optics 186 (2019) 288–292
A. Biswas, et al.
model followed by the study of soliton perturbation theory. These results are awaited currently but will be visible with time. Conflicts of interest The authors also declare that there is no conflict of interest. Acknowledgements The research work of the fourth author (MRB) was supported by the grant NPRP 8-028-1-001 from QNRF and he is thankful for it. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with Kerr law nonlinearity by F-expansion, Optik 181 (2019) 1028–1038. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with quadratic–cubic law by F-expansion, Optik 182 (2019) 930–943. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with cubic–quintic–septic law by F-expansion, Optik 182 (2019) 897–906. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with non-local nonlinearity by F-expansion, Optik 183 (2019) 1140–1150. A. Biswas, J.V. Guzman, M.F. Mahmood, S. Khan, M. Ekici, Q. Zhou, S.P. Moshokoa, M.R. Belic, Highly dispersive optical solitons with undetermined coefficients, Optik 182 (2019) 890–896. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with Kerr law nonlinearity by extended Jacobi's elliptic function expansion, Optik 183 (2019) 395–400. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with cubic–quintic–septic law by extended Jacobi's elliptic function expansion, Optik 183 (2019) 571–578. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with non-local nonlinearity by extended Jacobi's elliptic function expansion, Optik 184 (2019) 277–286. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with quadratic–cubic nonlinearity by exp-function, Optik (2019) in press. A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Highly dispersive optical solitons with cubic–quintic–septic law by exp-function, Optik (2019) in press. M. Ekici, M. Mirzazadeh, A. Sonmezoglu, Q. Zhou, H. Triki, M.Z. Ullah, S.P. Moshokoa, A. Biswas, Optical solitons in birefringent fibers with Kerr nonlinearity by exp-function method, Optik 131 (2017) 964–976. F. Ferdous, M.G. Hafez, A. Biswas, M. Ekici, Q. Zhou, M. Alfiras, S.P. Moshokoa, M. Belic, Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized exp(−ϕ(ξ))-expansion, Optik 178 (2019) 439–448. N. Kadkhoda, H. Jafari, Analytical solutions of the Gerdjikov–Ivanov equation by using exp(−ϕ(ξ))-expansion method, Optik 139 (2017) 72–76. K. Khan, M.A. Akbar, Application of exp(−Φ(ξ )) -expansion method to find the exact solutions of modified Benjamin–Bona–Mahony equation, World Appl. Sci. J. 24 (10) (2013) 1373–1377. H.O. Roshid, M.R. Kabir, R.C. Bhowmik, B.K. Datta, Investigation of solitary wave solutions for Vakhnenko–Parkes equation via exp-function and exp(−Φ(ξ )) -expansion method, Springer Plus 3 (2014) 692.
292
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Highly dispersive optical solitons with non-local nonlinearity by exp-function
T
⁎
Anjan Biswasa,b,c, Mehmet Ekicid, , Abdullah Sonmezoglud, Milivoj R. Belice a
Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762-7500, USA Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa d Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey e Institute of Physics Belgrade, Pregrevica 118, 11080 Zemun, Serbia b c
A R T IC LE I N F O
ABS TRA CT
OCIS: 060.2310 060.4510 060.5530 190.3270 190.4370Keywords: Dispersive solitons Non-local nonlinearity exp−function
This paper recovers singular and bright-singular highly dispersive optical solitons, having nonlocal form of nonlinearity, by the application of exp-function integration scheme. The existence criteria for these solitons are also listed in the work.
1. Introduction The advancement of optical solitons is primarily due to deep mathematical theories that are applicable to study this dynamics. One of the aspects of soliton dynamics is highly dispersive optical solitons which appears when inter-modal dispersion (IMD), thirdorder dispersion (3OD), fourth-order dispersion (4OD), fifth-order dispersion (5OD) and sixth-order dispersion (6OD) terms are taken into account in addition to the presence of the usual group-velocity dispersion (GVD) [1–10]. Several mathematical techniques have been applied to handle such highly dispersive optical solitons. These are the method of undetermined coefficients, F-expansion scheme, extended Jacobi's elliptic function expansion and lately the exp-function method. While the first method turned out to be an epic failure to retrieve soliton solutions [5], the second and third methods have proved to be a grand success in retrieving soliton solutions to the governing nonlinear Schrödinger's equation (NLSE) with six dispersion terms [1–4,6–8]. Finally, the exp-function scheme has proved to be successful with quadratic–cubic law and cubic–quintic–septic law of nonlinearity, thus far [9,10]. It has also been successfully applied to other physical systems such as birefringent fibers, water waves, oblique solitons and other such [11–15]. This paper therefore encouragingly studies highly dispersive optical solitons with non-local law of nonlinearity using exp-function. The striking details are presented in the rest of the paper after an intro to the governing model. 1.1. Governing model The dimensionless form of NLSE with non-local nonlinearity in presence of dispersion terms of all orders is [1–10]:
⁎
Corresponding author. E-mail address: [email protected] (M. Ekici).
https://doi.org/10.1016/j.ijleo.2019.04.082 Received 18 February 2019; Accepted 16 April 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 186 (2019) 288–292
A. Biswas, et al.
iq t + ia1 qx + a2 qxx + ia3qxxx + a4 qxxxx + ia5qxxxxx + a6 qxxxxxx + b (|q|2 ) xx q = 0.
(1)
Here, in (1), q(x, t) represents soliton molecules where x and t are independent spatial and temporal variables respectively. The first term represents linear temporal evolution, while a1, a2, a3, a4, a5 and a6 are coefficients of IMD, GVD, 3OD, 4OD, 5OD and 6OD respectively. Finally, b is the coefficient of non–local nonlinearity. The complex–valued function q(x, t) is the wave profile that represents the dependent variable. The coefficients are all real–valued constants and i = −1 . 2. Preliminaries To start off, the starting point is the hypothesis given by
q (x , t ) = g (ς ) eiϕ (x , t )
(2)
ς = x − vt
(3)
where
with v denotes the velocity of the soliton. Next, the phase component is defined as
ϕ (x , t ) = −κx + ωt + θ
(4)
where κ is the soliton frequency, ω is its wave number and θ is the phase constant. Once putting (2) into (1) real and imaginary parts yield
− (ω + κ (κ (a2 + κ (a3 + a6 κ 3 − κ (a4 + a5 κ ))) − a1 )) g + 2b g(g ′)2 + (a2 + κ (3a3 + κ (5κ (3a6 κ − 2a5) − 6a4 ))) g ″ + 2bg 2g ″ + (a4 + 5κ (a5 − 3a6 κ )) g (4) + a6 g (6) = 0
(5)
(v − a1 + 2a2 κ + 3a3 κ 2 − 4a4 κ 3 − 5a5 κ 4 + 6a6 κ 5) g + (−a3 + 2κ (2a4 + 5κ (a5 − 2a6 κ ))) g ″ − (a5 − 6a6 κ ) g (4) = 0
(6)
and
2
2
respectively. The notations g′ = dg/dς, g″ = d g/dς and so on are introduced. From (6), the constraint conditions are (7)
a5 = 6a6 κ a3 =
4κ (3a4 + 5a5 κ ) 3
(8)
and thus the soliton speed comes up as below:
v = a1 − 2κ (a2 + 4a4 κ 2 + 8a5 κ 3).
(9)
In view of (7) and (8), the real part can be rewritten as
− κ (6ω − 6a1 κ + 6a2 κ 2 + 18a4 κ 4 + 35a5 κ 5) g + 12bκg (g ′)2 + 3κ (2a2 + κ 2 (12a4 + 25a5 κ )) g ″ + 12bκg 2g ″ + 3κ (2a4 + 5a5 κ ) g (4) + a5 g (6) = 0.
(10)
3. exp(−ψ (ς ))−expansion scheme In order to investigate highly dispersive optical solitons to the NLSE with non-local nonlinearity by exp(−ψ (ς )) -expansion [11–15], the considered hypothesis is: N
g (ς ) =
∑ γj (exp[−ψ (ς )]) j .
(11)
j=1
Here, in (11), the coefficients γj are constants to be fixed later, such that γN ≠ 0, and the function ψ(ς) holds
ψ′ (ς ) = exp[−ψ (ς )] + χ exp[ψ (ς )] + ϑ
(12)
where the solutions of (12) are: If χ ≠ 0 and ϑ2 − 4χ > 0, 2
ϑ − 4χ ⎡ ϑ2 − 4χ tanh ⎛ 2 (ς + c ) ⎞ + ϑ ⎤ ⎢ ⎥ ⎝ ⎠ ψ (ς ) = ln ⎢− ⎥. 2χ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
(13)
For χ ≠ 0 and ϑ − 4χ < 0, 2
289
Optik - International Journal for Light and Electron Optics 186 (2019) 288–292
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2
⎡ 4χ − ϑ2 tan ⎛ 4χ − ϑ (ς + c ) ⎞ − ϑ ⎤ 2 ⎥ ⎢ ⎝ ⎠ ψ (ς ) = ln ⎢ ⎥. 2 χ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣
(14)
When χ = 0, ϑ ≠ 0 and ϑ2 − 4χ > 0,
ϑ ⎤. ψ (ς ) = −ln ⎡ ⎢ ⎣ exp(ϑ(ς + c )) − 1 ⎥ ⎦
(15)
Whenever χ ≠ 0, ϑ ≠ 0 and ϑ − 4χ = 0, 2
ψ (ς ) = ln ⎡− ⎢ ⎣
2(ϑ(ς + c ) + 2) ⎤ . ⎥ ϑ2 (ς + c ) ⎦
(16)
However, if χ = 0, ϑ = 0 and ϑ − 4χ = 0, 2
ψ (ς ) = ln[ς + c ].
(17) 2
It is important to note that c is the integration constant. Balancing g g″ with g (10) is of the form
(6)
in (10) gives N = 2. Therefore, the solution of Eq.
g (ς ) = γ0 + γ1 exp[−ψ (ς )] + γ2 exp[−2ψ (ς )].
(18)
Inserting (18) into (10), combining up the coefficients of exp(−ψ (ς )) , and coping with the resulting system, one gets the following sets: Set-1:
γ0 = γ2 χ , γ1 = γ2 ϑ, γ2 = γ2, 30a1 κ 2 − a5 (25κ 6 + 84κ 4 (ϑ2 − 4χ ) + 35κ 2 (ϑ2 − 4χ )2 + 12(4χ − ϑ2)3) , 30κ 4 2 2 2 2 a (75κ + 168κ (ϑ − 4χ ) + 35(ϑ − 4χ ) ) a2 = 5 , 30κ a (75κ 2 + 28(ϑ2 − 4χ )) , a4 = − 5 30κ 42a5 b=− 2 , γ2 κ ω=
(19)
Set-2:
γ2 (3χ + ϑ2) , 7 γ1 = γ2 ϑ, γ2 = γ2,
γ0 =
κ (42a1 − a5 (35κ 4 + 84κ 2 (4χ − ϑ2) + 37(ϑ2 − 4χ )2)) , 42 a (105κ 4 + 168κ 2 (4χ − ϑ2) + 37(ϑ2 − 4χ )2) a2 = 5 , 42κ 2 2 a (15κ + 16χ − 4ϑ ) , a4 = − 5 6κ 42a b = − 2 5. γ2 κ ω=
(20)
By employing the first set of parameters, one can secure bright-singular combo solitons solution
γ2 χ (4χ − ϑ2)
q (x , t )= ⎛⎜ϑ cosh ⎡ ⎣ ⎝
ϑ2 − 4χ (c + ς ) 2
⎤+ ⎦
ϑ2 − 4χ sinh ⎡ ⎣
ϑ2 − 4χ (c + ς ) 2
2
⎤ ⎟⎞ ⎦⎠
2 6 4 2 2 2 2 2 3 ⎧ ⎛ 30a1 κ − a5 (25κ + 84κ (ϑ − 4χ ) + 35κ (ϑ − 4χ ) + 12(4χ − ϑ ) ) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 30 κ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
trigonometric function solution 290
(21)
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A. Biswas, et al.
γ2 χ (4χ − ϑ2)
q (x , t )= ⎛⎜ϑ cos ⎡ ⎣ ⎝
4χ − ϑ2 (c + ς ) 2
⎤− ⎦
4χ − ϑ2 sin ⎡ ⎣
4χ − ϑ2 (c + ς ) 2
2
⎤ ⎟⎞ ⎦⎠
2 6 4 2 2 2 2 2 3 ⎧ ⎛ 30a1 κ − a5 (25κ + 84κ (ϑ − 4χ ) + 35κ (ϑ − 4χ ) + 12(4χ − ϑ ) ) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 30 κ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(22)
singular soliton solution
q (x , t ) =
2 6 4 2 2 4 6 γ2 ϑ2 ϑ(c + ς ) ⎤ ⎧ ⎛ 30a1 κ − a5 (25κ + 84κ ϑ + 35κ ϑ − 12ϑ ) ⎞ t + θ ⎫ ⎤ exp ⎡ csch2 ⎡ ⎢i ⎨−κx + ⎥ ⎬ 2 30 4 κ ⎣ ⎦ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(23)
and finally plane wave solutions
q (x , t ) =
2 6 γ2 ⎛ ϑ3 (c + ς )[ϑ(c + ς ) + 4] ⎞ ⎡i ⎧−κx + ⎛ 30a1 κ − 25a5 κ ⎞ t + θ ⎫ ⎤ ⎜4χ − ⎟ exp 2 ⎢ ⎬⎥ [ϑ(c + ς ) + 2] 30κ 4⎝ ⎝ ⎠ ⎠ ⎩ ⎭ ⎦ ⎣ ⎨ ⎜
⎟
(24)
2 6 ⎧ ⎛ 30a1 κ − 25a5 κ ⎞ t + θ ⎫ ⎤. q (x , t ) = γ2 (c + ς )2exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 30 κ ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(25)
Next, by the use of the second set of parameters, one can explore singular soliton solution
⎡ γ2 ⎢ 2 ⎢ϑ + q (x , t )= 7⎢ ⎛⎜ϑ + ⎢ ⎝ ⎣
⎛ ⎜ +χ 3− 2 ⎜ ϑ2 − 4χ (c + ς ) ⎞ 2 ⎡ ⎤ ϑ+ ϑ − 4χ tanh ⎟ ⎜ 2 ⎝ ⎣ ⎦⎠ 28χ 2
⎞⎤ ⎥ ⎟⎥ ϑ2 − 4χ (c + ς ) ⎟ ⎥ ⎤⎟ ϑ2 − 4χ tanh ⎡ ⎥ 2 ⎣ ⎦⎠⎦ 14ϑ
4 2 2 2 2 ⎧ ⎛ κ (42a1 − a5 (35κ + 84κ (4χ − ϑ ) + 37(ϑ − 4χ ) )) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(26)
periodic solution
⎡ γ2 ⎢ 2 ⎢ϑ + q (x , t )= 7⎢ ⎛⎜ϑ − ⎢ ⎝ ⎣
⎛ ⎜ +χ 3− 2 ⎜ 4χ − ϑ2 (c + ς ) ⎞ 2 ⎤⎟ ϑ− 4χ − ϑ tan ⎡ ⎜ 2 ⎝ ⎣ ⎦⎠ 28χ 2
⎞⎤ ⎥ ⎟⎥ 2 ⎟ 4χ − ϑ (c + ς ) ⎤⎟⎥ 4χ − ϑ2 tan ⎡ ⎥ 2 ⎣ ⎦⎠⎦ 14ϑ
4 2 2 2 2 ⎧ ⎛ κ (42a1 − a5 (35κ + 84κ (4χ − ϑ ) + 37(ϑ − 4χ ) )) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎥ ⎢i ⎨−κx + ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(27)
bright-singular combo solitons solution
1 1 1 ⎞ + ⎟ q (x , t )= γ2 ϑ2 ⎛⎜ + cosh[ϑ(c + ς )] + sinh[ϑ(c + ς )] − 1 (cosh[ϑ(c + ς )] + sinh[ϑ(c + ς )] − 1)2 ⎠ ⎝7 4 2 2 4 ⎧ ⎛ κ (42a1 − a5 (35κ − 84κ ϑ + 37ϑ )) ⎞ t + θ ⎫ ⎤ × exp ⎡ ⎢i ⎨−κx + ⎥ ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(28)
and finally plane waves
q (x , t ) =
γ2 (12χ [ϑ(c + ς ) + 2]2 + ϑ2 (16 − 3ϑ(c + ς )[ϑ(c + ς ) + 4])) κ (42a1 − 35a5 κ 4 ) ⎞ exp ⎡ i ⎧−κx + ⎛ t + θ⎫ ⎤ 2 ⎢ ⎬⎥ ⎨ 28[ϑ(c + ς ) + 2] 42 ⎝ ⎠ ⎩ ⎭ ⎦ ⎣ ⎜
4 ⎧ ⎛ κ (42a1 − 35a5 κ ) ⎞ t + θ ⎫ ⎤. q (x , t ) = γ2 (c + ς )2exp ⎡ ⎢i ⎨−κx + ⎥ ⎬ 42 ⎝ ⎠ ⎭⎦ ⎣ ⎩ ⎜
⎟
(29)
⎟
(30)
4. Conclusions This paper recovered highly dispersive optical solitons having non-local nonlinearity by exp-function. It is only singular and bright-singular combo solitons that emerged from this scheme. Surely, this algorithm fails to retrieve the necessary and important bright and dark soliton solutions to the model. Therefore, in order to get a complete spectrum of soliton solutions to the model, one must resort to additional integration algorithms such as Lie symmetry analysis, Kudryashov's scheme, G′/G-expansion scheme and several others. It is expected that these mechanisms will achieve our goal. Later on, the conservation laws will also be derived for the 291
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model followed by the study of soliton perturbation theory. These results are awaited currently but will be visible with time. Conflicts of interest The authors also declare that there is no conflict of interest. Acknowledgements The research work of the fourth author (MRB) was supported by the grant NPRP 8-028-1-001 from QNRF and he is thankful for it. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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