- Email: [email protected]
Optical solitons of Gerdjikov–Ivanov equation in birefringent fibers with modified simple equation scheme
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons of Gerdjikov–Ivanov equation in birefringent fibers with modified simple equation scheme
T
Yakup Yildirim Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: Birefringent fibers The coupled Gerdjikov–Ivanov model Modified simple equation algorithm Optical solitons
The coupled system corresponding to Gerdjikov–Ivanov equation is employed to obtain optical solitons of its in this current study. The modified simple equation integration procedure is used to get not only such strategic solitons namely dark, bright with singular solitons but also singular periodic solutions.
1. Introduction Solitons in optics create the basic fabric for soliton transmission technology along with transcontinental and transoceanic distances, optical fibers, data transmission across, telecommunications industry that are constituted through the medium of a lot of models [1–19]. Gerdjikov–Ivanov (GI) model, which has achieved popularity especially in not only photonic crystal fibers but also nonlinear fiber optics since its first appearance, is one of these governing models and furthermore generally has been studied in polarization preserving fibers along with strategic algorithms that are the coupled amplitude-phase methodology, modified simple equation technique, trial equation technique, extended trial equation procedure, extended Kudryashov's method, the extended tanh–coth procedure, semi-inverse variational methodology, the csch scheme, the sine–cosine approach and also G′/G2-expansion technique [1–10]. In despite of these advancements, the solitons along the only one component of the model have been taken into consideration. In order to improve further the model by making contributions to all these studies, the model has been given in two component form for vector solitons with and without four-wave mixing terms (FWM) [11,12] which give rise to improve the model further along. Because of all these reasons, the GI equation without FWM in birefringent fibers, which has an important place to govern parallel transmission of pulses through optical fibers, will be revisited and is going to be investigated along with modified simple equation methodology which recovers strategic solitons namely dark, bright with singular solitons with the inclusion of the existence of circumstances for these important solitons. Additionally, singular-periodic solutions are going to be emerged as a result of the reverse form of the restrictions. In the advancing sections of this work, the details of the analysis of the GI model are given. 1.1. Governing model The Gerdjikov–Ivanov equation [1–10] is imparted by
iψt + aψxx + b|ψ|4 ψ + icψ2ψx* = 0.
(1)
The first term is referred to as the temporal evolution of pulses when the existence of group velocity dispersion is supplied by the coefficient of a in this quite important governing model. The complex valued function ψ(x, t) is referred to as the wave profile. The
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.01.047 Received 19 January 2019; Accepted 19 January 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
coefficient of b is named as the nonlinear term that signifies quintic nonlinearity. Once and for all, the existence of a form of dispersive phenomenon is ensured with the coefficient of c. The Gerdjikov–Ivanov model without FWM in birefringent fibers [11] is described by
iψt + a1 ψxx + (ξ1 |ψ|4 + η1 |ψ|2 |ϕ|2 + ζ1 |ϕ|4 ) ψ + i (β1 ψ2 + γ1 ϕ2) ψx* = 0, iϕt + a2 ϕxx + (ξ2 |ϕ|4 + η2 |ϕ|2 |ψ|2 + ζ2 |ψ|4 ) ϕ + i (β2 ϕ2 + γ2 ψ2) ϕx* = 0.
(2)
The coefficients of ai correspond to group velocity dispersion when the coefficients of ξi stem from self-phase modulation in this coupled GI system. Once and for all, the coefficients of ηi as well as ζi correspond to cross-phase modulation whilst the coefficients of βi, γi account for other forms of dispersive phenomenon along with i = 1, 2. 2. Pen-picture of modified simple equation procedure Succinct overview of modified simple equation algorithm [13–19] is going to be imparted before its feasibility to the coupled GI system in order to locate strategic optical solitons. Step-1: First of all, we will employ a NLEE in general form
Ψ(ζ , ζt , ζ x , ζtt, ζ xt, ζ xx , …) = 0
(3)
and later on inserting the wave variable transformation which is a very important place in this method
ζ (x , t ) = Θ(ϑ)
(4)
along with (5)
ϑ = x − vt into Eq. (3) give rise to
Φ(Θ, Θ′, Θ′ ′, Θ′ ′′, …) = 0
(6)
which is the reduced equation and also known as ordinary differential equation. In (6), the function Θ = Θ(ξ) is referred to as an dependent function and furthermore Φ corresponds to a polynomial with the involving of the dependent function Θ and its derivatives. Step-2: The auxiliary equation that is another key point of this method is going to be addressed by means of N
Θ(ϑ) =
i
Q′ (ϑ)
∑ δi ⎛ Q (ϑ) ⎞ i=0
⎜
⎟
⎝
⎠
(7)
which involves the required coefficients δ0, δ1, …, δN and this coefficients have to be evaluated to get optical solitons. Step-3: The existence of the N number as a result of the application of the balancing principle is vital in the use of the auxiliary equation. Step-4: An overdeterminet equation system are occurred under the condition of inserting Eq. (7) into Eq. (6) and later on putting together the coefficient of the same functions Q−i(ϑ), i = 1, 2, … to zero. As a result of the solution of the system, the required coefficients and Q(ϑ) are granted. 2.1. Soliton solutions The modified simple equation technique, that will be employed on account of locating exact solutions to Eq. (2), is known as one of the most strategic algorithms in literature. In order to get important optical solitons of the coupled GI system, the following solution condition
ψ (x , t ) = P1 (ϑ) eiφ (x , t ), ϕ (x , t ) = P2 (ϑ) eiφ (x , t )
(8)
is going to be taken into consideration. With the inclusion of the parameter
ϑ = k (x − ρt )
(9)
which includes the speed of the soliton ρ, the function Pj(ϑ) accounts for the amplitude component while the function φ(x, t) corresponds to the phase component and also this function will be given as
φ (x , t ) = −κx + ωt + θ .
(10)
In Eq. (10), the parameter κ is referred to as soliton frequency while the parameter ω corresponds to soliton wave number and lastly the parameter θ implies to soliton phase. The real component and the imaginary component are granted by means of
aj k 2P′j ′ − (ω + aj κ 2) Pj − κβj P 3j − κγj Pj P j2˜ + ξ j P 5j + ηj P 3j P j2˜ + ζ j Pj P j˜4 = 0 425
(11)
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
and
− ρ − 2aj κ + βj P j2 + γj P j2˜ = 0
(12)
sequentially through the medium of inserting Eq. (8) into Eq. (2) for j = 1, 2 and j˜ = 3 − j . The following condition easily can be get
P j˜ = Pj
(13)
on account of implement the balancing rule in (11) and this condition convert the real part (11) and the imaginary part (12) into
aj k 2P′j ′ − (ω + aj κ 2) Pj − κ (βj + γj ) P 3j + (ξ j + ηj + ζ j ) P 5j = 0,
(14)
ρ = −2aj κ + (βj + γj ) P j2
(15)
respectively. The speed of the soliton is yield along with Eq. (15) while the profile of the soliton is emerged under the condition of integrating Eq. (14). Case-1: Comparing P′j ′ between P 3j in Eq. (14), one can attain N = 1 so the auxiliary equation (7) is imparted by
Q′ (ϑ) ⎞ Pj (ϑ) = δ0 + δ1 ⎛ . ⎝ Q (ϑ) ⎠ ⎜
⎟
(16)
The following overdeterminet equations can be imparted by means of Q−5 coeff.:
δ15 (Q′)5 (ξ j + ηj + ζ j ) = 0,
(17)
Q−4 coeff.:
5δ0 δ14 (Q′) 4 (ξ j + ηj + ζ j ) = 0, Q
−3
(18)
coeff.:
δ1 (Q′)3 (10δ02 δ12 ηj + 10δ02 δ12 ξ j + 10δ02 δ12 ζ j − κβj δ12 − κδ12 γj + 2k 2aj ) = 0, Q
−2
(19)
coeff.:
− δ1 Q′ ((−10δ03 δ1 ηj − 10δ03 δ1 ξ j − 10δ03 δ1 ζ j + 3κβj δ0 δ1 + 3κδ0 δ1 γj ) Q′ + 3aj k 2Q′ ′) = 0, Q
−1
(20)
coeff.:
− δ1 ((−5δ04 ηj − 5δ04 ξ j − 5δ04 ζ j + 3κβj δ02 + 3κδ02 γj + κ 2aj + ω) Q′ − k 2aj Q′ ′′) = 0,
(21)
Q0 coeff.:
δ0 (δ04 ηj + δ04 ξ j + δ04 ζ j − κβj δ02 − κγj δ02 − κ 2aj − ω) = 0,
(22)
under the condition of inserting Eq. (16) into Eq. (14) and later on putting together the coefficient of the same functions to zero. As a result of the solution of the system (17)–(22), the required coefficients are granted as follows
δ0 = ± −
ω + aj κ 2 κ (βj + γj )
,
δ1 = ±
2aj k 2 κ (βj + γj )
,
ηj = −ξ j − ζ j
and
Q′ ′ = ± −
Q′ ′′ = −
2(ω + aj κ 2)
Q′,
aj k 2
(23)
2(ω + aj κ 2) aj k 2
Q′.
(24)
One can easily reach out to
Q′ = ± −
aj k 2 2(ω + aj
± −
κ 2)
k1 e
2(ω + aj κ 2) aj k 2
ϑ
, (25)
and 426
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
Q=−
aj k 2 2(ω + aj
± −
κ 2)
k1 e
2(ω + aj κ 2)
ϑ
aj k 2
+ k2,
(26)
by means of using Eqs. (23) and (24). On account of inserting Eqs. (25) and (26) which involve k1 and k2 integration constants into Eq. (16), the exact solutions to Eq. (2) are granted by means of
ψ (x , t ) =⎧± ⎨ ⎩
ω + a1 κ 2 1 + γ1)
− κ (β
±
2a1 k2 κ (β1 + γ1)
2
2(ω + a1 κ ) ⎛ k (x − ρt ) ⎞ ⎫ ± − a1 k2 a1 k 2 ⎜ ± − 2(ω + a1κ2) k1 e ⎟ ⎪ i (−κx + ωt + θ) ×⎜ ⎟⎬e 2(ω + a1 κ 2) k (x − ρt ) ± − ⎜ − a1k2 k1 e ⎟⎪ a1 k 2 + k 2 2 ⎝ 2(ω + a1κ ) ⎠⎭
and
ϕ (x , t ) =⎧± ⎨ ⎩
ω + a2 κ 2 2 + γ2)
− κ (β
±
2a2 k2 κ (β2 + γ2)
2
2(ω + a2 κ ) ⎛ k (x − ρt ) ⎞ ⎫ ± − a2 κ 2 a2 k 2 ± − k e 1 ⎟ ⎪ i (−κx + ωt + θ) ⎜ 2(ω + a2 κ 2) ×⎜ ⎟⎬e 2(ω + a2 κ 2) k (x − ρt ) ± − ⎜ − a2 κ2 k1 e a2 k 2 + k2 ⎟ ⎪ 2 ⎠⎭ ⎝ 2(ω + a2 κ )
If we set
k1 = −
2(ω + aj κ 2) aj κ 2
± −
e
2(ω + aj κ 2) aj k 2
ϑ0
,
k2 = ± 1,
we get:
ψ (x , t ) = ± −
ϕ (x , t ) = ± −
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ ) , tanh ⎡ (k (x − ρt ) + ϑ0) ⎤ − 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(27)
ω + a2 κ 2 ω + a2 κ 2 ⎤ i (−κx + ωt + θ), tanh ⎡ ⎢ − 2a2 k 2 (k (x − ρt ) + ϑ0) ⎥ e κ (β2 + γ2) ⎣ ⎦
(28)
where Eqs. (27) and (28) means dark soliton solutions whenever
aj (ω + aj κ 2) < 0. ψ (x , t ) = ± −
ϕ (x , t ) = ± −
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ) , coth ⎡ (k (x − ρt ) + ϑ0) ⎤ − 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(29)
ω + a2 κ 2 ω + a2 κ 2 ⎤ i (−κx + ωt + θ). coth ⎡ ⎢ − 2a2 k 2 (k (x − ρt ) + ϑ0) ⎥ e κ (β2 + γ2) ⎣ ⎦
(30)
These solutions stem from singular soliton solutions under the condition of
aj (ω + aj κ 2) < 0. ψ (x , t ) = ±
ϕ (x , t ) = ±
ψ (x , t ) = ±
ϕ (x , t ) = ±
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ ) , tan ⎡ (k (x − ρt ) + ϑ0) ⎤ 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(31)
ω + a2 κ 2 ω + a2 κ 2 i (−κx + ωt + θ) , tan ⎡ (k (x − ρt ) + ϑ0) ⎤ ⎢ ⎥e κ (β2 + γ2) 2a2 k 2 ⎣ ⎦
(32)
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ ) , cot ⎡ (k (x − ρt ) + ϑ0) ⎤ 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(33)
ω + a2 κ 2 ω + a2 κ 2 i (−κx + ωt + θ) . cot ⎡ (k (x − ρt ) + ϑ0) ⎤ ⎢ ⎥e κ (β2 + γ2) 2a2 k 2 ⎣ ⎦
(34)
427
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
These solutions correspond to singular periodic solutions as long as
aj (ω + aj κ 2) > 0. Case-2: Eq. (14) can easily be turned into the following form
aj k 2 (−(V ′j )2 + 2Vj V ′j ′) − 4(ω + aj κ 2) V j2 − 4κ (βj + γj ) V j3 + 4(ξ j + ηj + ζ j ) V j4 = 0, by means of getting Pj = imparted by
1 V j2 .
Comparing (V ′j )2 or Vj V ′j ′ between V j4 in Eq. (35), one ensure N = 1 so the auxiliary equation (7) is
Q′ (ϑ) ⎞ Vj (ϑ) = δ0 + δ1 ⎛ . ⎝ Q (ϑ) ⎠ ⎜
(35)
⎟
(36)
The following overdeterminet equations can be imparted by means of Q−4 coeff.:
(Q′) 4δ12 (3k 2aj + 4δ12 ηj + 4δ12 ξ j + 4δ12 ζ j ) = 0, Q
−3
(37)
coeff.:
4δ1 (Q′)2 ((k 2aj δ0 − κβj δ12 − κδ12 γj + 4δ0 δ12 ηj + 4δ0 δ12 ξ j + 4δ0 δ12 ζ j ) Q′ − k 2aj δ1 Q′ ′) = 0,
(38)
Q−2 coeff.:
δ1 ((−4κ 2aj δ1 − 12κβj δ0 δ1 − 12κδ0 δ1 γj + 24δ02 δ1 ηj + 24δ02 δ1 ξ j + 24δ02 δ1 ζ j − 4ωδ1)(Q′)2 − 6k 2aj δ0 Q′Q′ ′ + 2k 2aj δ1 Q′Q′ ′′ − k 2aj δ1 (Q′ ′)2) = 0, Q
−1
(39)
coeff.:
2δ0 δ1 ((−4κ 2aj − 6κβj δ0 − 6κδ0 γj + 8δ02 ηj + 8δ02 ξ j + 8δ02 ζ j − 4ω) Q′ + 3k 2aj Q′ ′′) = 0,
(40)
0
Q coeff.:
4δ02 (−κ 2aj − κβj δ0 − κδ0 γj + δ02 ηj + δ02 ξ j + δ02 ζ j − ω) = 0,
(41)
under the condition of inserting Eq. (36) into Eq. (35) and later on putting together the coefficient of the same functions to zero. As a result of the solution of the system (37)–(41), the required coefficients are granted as follows
δ0 =
3κ (βj + γj ) 4(ξ j + ηj + ζ j )
ω=−
,
δ1 = ± −
3aj k 2 4(ξ j + ηj + ζ j )
,
κ 2 (16aj ηj + 16aj ξ j + 16aj ζ j + 3βj2 + 6βj γj + 3γ j2) 16(ξ j + ηj + ζ j )
and
Q′ ′ = ± −
Q′ ′′ = −
3κ 2 (βj + γj )2
Q′,
4aj k 2 (ξ j + ηj + ζ j ) 3κ 2 (βj + γj )2
4aj k 2 (ξ j + ηj + ζ j )
(42)
Q′. (43)
One can easily reach out to
Q′ = ± −
4aj k 2 (ξ j + ηj + ζ j ) 3κ 2 (βj
+ γj
)2
± −
k1 e
3κ 2 (βj + γj )2 4aj k 2 (ξ j + ηj + ζ j )
ϑ
, (44)
and
Q=−
4aj k 2 (ξ j + ηj + ζ j ) 3κ 2 (βj
+ γj
)2
± −
k1 e
3κ 2 (βj + γj )2 4aj k 2 (ξ j + ηj + ζ j )
ϑ
+ k2
(45)
by means of using Eqs. (42) and (43). On account of inserting Eqs.(44) and (45) which involve k1 and k2 integration constants into Eq. 428
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
(36), the exact solutions to Eq. (2) are granted by means of 3κ (β + γ )
ψ (x , t ) =⎧ 4(ξ +1η +1ζ ) ± ⎨ ⎩ 1 1 1
3a1 k2
− 4(ξ
1 + η1 + ζ1)
2
2
3κ (β1+ γ1) ⎛ ± − k (x − ρt ) ⎞ ⎫ 4a1 k2 (ξ1 + η1 + ζ1) 4a1 k 2 (ξ1+ η1+ ζ1) ± − k e 1 ⎟⎪ ⎜ ⎪ i (−κx + ωt + θ) 3κ 2 (β1 + γ1)2 ×⎜ ⎟⎬e 2 2 3κ (β1+ γ1) ⎟⎪ ⎜ 4a1k2 (ξ1 + η1 + ζ1) ± − 4a1k 2 (ξ + η + ζ ) k (x − ρt ) 1 1 1 k1 e + k2 ⎟ ⎜− 2 2 ⎪ ⎠⎭ ⎝ 3κ (β1 + γ1)
and 3κ (β + γ )
ϕ (x , t ) =⎧ 4(ξ +2η +2ζ ) ± ⎨ ⎩ 2 2 2
− 4(ξ
3a2 k2
2 + η2 + ζ2)
2
2
3κ (β2 + γ2) ⎛ ± − k (x − ρt ) ⎞ ⎫ 4a2 k2 (ξ2 + η2 + ζ2) 4a2 k 2 (ξ2+ η2 + ζ2) ± − k e 1 ⎟⎪ ⎜ ⎪ i (−κx + ωt + θ) 3κ 2 (β2 + γ2)2 ×⎜ ⎟⎬e 2 2 3κ (β2 + γ2) ⎟⎪ ⎜ 4a2 k2 (ξ2 + η2 + ζ2) ± − 4a2 k 2 (ξ + η + ζ ) k (x − ρt ) 2 2 2 k1 e + k2 ⎟ ⎜− 2 2 ⎪ ⎠⎭ ⎝ 3κ (β2 + γ2)
If we set
k1 = −
3κ 2 (βj + γj )2 4aj
k 2 (ξ
± −
+ ηj + ζ j )
j
e
3κ 2 (βj + γj )2 4aj k 2 (ξ j + ηj + ζ j )
ϑ0
,
k2 = ± 1,
we get: 3κ (β + γ ) 3κ 2 (β1 + γ1)2 ψ (x , t ) = ⎧ 8(ξ +1η +1ζ ) ⎜⎛1 ± tanh ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎟⎞ ⎫ 16aj k2 (ξ1 + η1 + ζ1) ⎥ ⎢ ⎨ 1 1 1 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
1 2
κ 2 (16a1 η1+ 16a1 ξ1+ 16a1 ζ1+ 3β12 + 6β1 γ1+ 3γ12) ⎛ ⎞ i ⎜−κx − t + θ⎟ 16(ξ1+ η1+ ζ1) ⎝ ⎠,
(46)
3κ (β + γ ) 3κ 2 (β2 + γ2)2 ϕ (x , t ) = ⎧ 8(ξ +2η +2ζ ) ⎛⎜1 ± tanh ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎟⎞ ⎫ 16aj k2 (ξ2 + η2 + ζ2) ⎥ ⎢ ⎨ 2 2 2 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
⎛ i ⎜−κx − ⎝
1 2
κ 2 (16a2 η2 + 16a2 ξ2+ 16a2 ζ2+ 3β22 + 6β2 γ2+ 3γ22) ⎞ t + θ⎟ 16(ξ2+ η2 + ζ2)
(47)
⎠.
These solutions stem from dark soliton solutions under the condition of
aj (ξ j + ηj + ζ j ) < 0. 3κ (β + γ ) 3κ 2 (β1 + γ1)2 ψ (x , t ) = ⎧ 8(ξ +1η +1ζ ) ⎜⎛1 ± coth ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎟⎞ ⎫ 16aj k2 (ξ1 + η1 + ζ1) ⎥ ⎢ ⎨ 1 1 1 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
⎛ i ⎜−κx − ⎝
1 2
κ 2 (16a1 η1+ 16a1 ξ1+ 16a1 ζ1+ 3β12 + 6β1 γ1+ 3γ12) ⎞ t + θ⎟ 16(ξ1+ η1+ ζ1)
⎠,
(48)
3κ (β + γ ) 3κ 2 (β2 + γ2)2 ϕ (x , t ) = ⎧ 8(ξ +2η +2ζ ) ⎛⎜1 ± coth ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎞⎟ ⎫ 16aj k2 (ξ2 + η2 + ζ2) ⎥ ⎢ ⎨ 2 2 2 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
⎛ i ⎜−κx − ⎝
1 2
κ 2 (16a2 η2 + 16a2 ξ2+ 16a2 ζ2+ 3β22 + 6β2 γ2+ 3γ22) ⎞ t + θ⎟ 16(ξ2+ η2 + ζ2)
⎠,
(49)
where Eqs. (48) and (49) singular soliton solutions whenever
aj (ξ j + ηj + ζ j ) < 0. Case-3: Comparing (V ′j )2 or Vj V ′j ′ between V 3j in Eq. (35), one ensure N = 2 so the auxiliary equation (7) is imparted by 2
Q′ (ϑ) ⎞ Q′ (ϑ) ⎞ Vj (ϑ) = δ0 + δ1 ⎛ + δ2 ⎛ . Q (ϑ) ⎝ ⎠ ⎝ Q (ϑ) ⎠ ⎜
⎟
⎜
⎟
(50)
The following overdeterminet equations can be imparted by means of 429
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
Q−8 coeff.:
4δ24 (Q′)8 (ηj + ξ j + ζ j ) = 0, Q
−7
(51)
coeff.:
16δ1 δ23 (Q′)7 (ηj + ξ j + ζ j ) = 0, Q
−6
(52)
coeff.:
4δ22 (Q′)6 (2k 2aj − κβj δ2 − κδ2 γj + 4δ0 δ2 ηj + 4δ0 δ2 ξ j + 4δ0 δ2 ζ j + 6δ12 ηj + 6δ12 ξ j + 6δ12 ζ j ) = 0, Q
−5
(53)
coeff.:
− 4δ2 (Q′) 4 ((−3k 2aj δ1 + 3κβj δ1 δ2 + 3κδ1 δ2 γj − 12δ0 δ1 δ2 ηj − 12δ0 δ1 δ2 ξ j − 12δ0 δ1 δ2 ζ j − 4δ13 ηj − 4δ13 ξ j − 4δ13 ζ j ) Q′ + 3k 2aj δ2 Q′ ′) = 0, Q
−4
(54)
coeff.:
− (Q′)3 ((−3k 2aj δ12 + 4δ22 aj κ 2 − 24ξ j δ02 δ22 − 24ηj δ02 δ22 − 24ζ j δ02 δ22 + 4δ22 ω − 4ξ j δ14 − 4ηj δ14 − 4ζ j δ14 − 12k 2aj δ0 δ2 + 12κβj δ0 δ22 + 12κβj δ12 δ2 + 12κγj δ0 δ22 + 12κγj δ12 δ2 − 48ξ j δ0 δ12 δ2 − 48ηj δ0 δ12 δ2 − 48ζ j δ0 δ12 δ2) Q′ + 18k 2aj δ1 δ2 Q′ ′ − 4k 2aj δ22 Q′ ′′) = 0, Q
−3
(55)
coeff.:
− 2(Q′)2 ((−2k 2aj δ0 δ1 + 4κ 2aj δ1 δ2 + 12κβj δ0 δ1 δ2 + 2κβj δ13 + 12κδ0 δ1 δ2 γj + 2κδ13 γj − 24δ02 δ1 δ2 ηj − 24δ02 δ1 δ2 ξ j − 24δ02 δ1 δ2 ζ j − 8δ0 δ13 ηj − 8δ0 δ13 ξ j − 8δ0 δ13 ζ j + 4ωδ1 δ2) Q′ + (10k 2aj δ0 δ2 + 2k 2aj δ12) Q′ ′ − 3k 2aj δ1 δ2 Q′ ′′) = 0,
(56)
Q−2 coeff.:
(−8κ 2aj δ0 δ2 − 4κ 2aj δ12 − 12κβj δ02 δ2 − 12κβj δ0 δ12 − 12κδ02 δ2 γj − 12κδ0 δ12 γj + 16δ03 δ2 ηj + 16δ03 δ2 ξ j + 16δ03 δ2 ζ j + 24δ02 δ12 ηj + 24δ02 δ12 ξ j + 24δ02 δ12 ζ j − 8ωδ0 δ2 − 4ωδ12)(Q′)2 − 6k 2aj δ0 δ1 Q′Q′ ′ + (4k 2aj δ0 δ2 + 2k 2aj δ12) Q′Q′ ′′ + (4k 2aj δ0 δ2 − k 2aj δ12)(Q′ ′)2 = 0, Q
−1
(57)
coeff.:
− 2δ0 δ1 ((4κ 2aj + 6κβj δ0 + 6κδ0 γj − 8δ02 ηj − 8δ02 ξ j − 8δ02 ζ j + 4ω) Q′ − k 2aj Q′ ′′) = 0,
(58)
0
Q coeff.:
− 4δ02 (κ 2aj + κβj δ0 + κδ0 γj − δ02 ηj − δ02 ξ j − δ02 ζ j + ω) = 0,
(59)
under the condition of inserting Eq. (50) into Eq. (35) and later on putting together the coefficient of the same functions to zero. As a result of the solution of the system (51)–(59), the required coefficients are granted as follows
δ0 = 0,
δ1 = ±
16k 2aj (ω + aj κ 2) κ 2 (βj + γj )2
,
δ2 =
2k 2aj κ (βj + γj )
,
ηj = −ξ j − ζ j
and
Q′ ′ = ±
Q′ ′′ =
4(ω + aj κ 2) aj k 2
4(ω + aj aj k 2
κ 2)
Q′, (60)
Q′.
(61)
One can easily reach out to 430
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
Q′ = ±
aj k 2
±
4(ω + aj
4(ω + aj κ 2)
k1 e
κ 2)
aj k 2
ϑ
, (62)
and
Q=
aj k 2
±
4(ω + aj κ 2)
k1 e
4(ω + aj κ 2)
ϑ
aj k 2
+ k2
(63)
by means of using Eqs. (60) and (61). On account of inserting Eqs. (62) and (63) which involve k1 and k2 integration constants into Eq. (50), the exact solutions to Eq. (2) are granted by means of 2
⎧ ⎪ ψ (x , t ) =
⎨ ⎪ ⎩
±
16k2a1 (ω + a1 κ 2) κ 2 (β1 + γ1)2
4(ω + a1 κ ) ⎛ k (x − ρt ) ⎞ ± a1 k2 a1 k 2 ⎟ ⎜ ± 4(ω + a1κ2) k1 e ⎟ ⎜ 4(ω + a1 κ 2) k (x − ρt ) ± ⎟ ⎜ a1k2 k1 e a1 k 2 + k 2 2 ⎠ ⎝ 4(ω + a1κ ) 2
1 2 2
⎫ ⎪ ⎪ i (−κx + ωt + θ) e ⎬ ⎪ ⎪ ⎭
4(ω + a1 κ ) ⎛ k (x − ρt ) ⎞ ± a1 k2 a1 k 2 ± k1 e ⎟ 2 ⎜ 2 4(ω + a1 κ ) 2k a1 + κ (β + γ ) ⎜ ⎟ 1 1 4(ω + a1 κ 2) k (x − ρt ) ± ⎜ a1k2 k1 e a1 k 2 + k2 ⎟ 2 ⎠ ⎝ 4(ω + a1κ )
and 2
⎧ ⎪ ϕ (x , t ) =
⎨ ⎪ ⎩
±
16k2a2 (ω + a2 κ 2) κ 2 (β2 + γ2)2
4(ω + a2 κ ) ⎛ k (x − ρt ) ⎞ ± a2 k2 a2 k 2 ⎟ ⎜ ± 4(ω + a2 κ2) k1 e ⎟ ⎜ 4(ω + a2 κ 2) ± k (x − ρt ) ⎜ a2 k2 k1 e a2 k 2 + k2 ⎟ 2 ⎠ ⎝ 4(ω + a2 κ ) 2
1 2 2
4(ω + a2 κ ) ⎛ ± k (x − ρt ) ⎞ a2 k2 a2 k 2 ± k1 e ⎟ ⎜ 2 4(ω + a2 κ ) 2k2a2 + κ (β + γ ) ⎜ ⎟ 2 2 4(ω + a2 κ 2) ± k (x − ρt ) ⎟ ⎜ a2 k2 k1 e a2 k 2 + k 2 2 ⎠ ⎝ 4(ω + a2 κ )
⎫ ⎪ ⎪ i (−κx + ωt + θ) e ⎬ ⎪ ⎪ ⎭
If we set
k1 =
4(ω + aj κ 2) aj
k2
±
e
4(ω + aj κ 2) aj k 2
ϑ0
,
k2 = ± 1,
we get: 1 2 2 ⎧ 2(ω + a1 κ 2) ⎡ ω + aj κ ⎤⎫ ψ (x , t ) = − sech2 ⎢ (k (x − ρt ) + ϑ0) ⎥ ei (−κx + ωt + θ), 2 ⎨ κ (β1 + γ1) ⎬ aj k ⎣ ⎦⎭ ⎩
(64)
1 2 ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) ⎫ i (−κx + ωt + θ) ϕ (x , t ) = − sech2 ⎡ (k (x − ρt ) + ϑ0) ⎤ e , 2 ⎢ ⎥ ⎨ κ (β2 + γ2) ⎬ a2 k ⎣ ⎦⎭ ⎩
(65)
where Eqs. (64) and (65) means bright soliton solutions whenever
aj (ω + aj κ 2) > 0. 1 2 2 ⎧ 2(ω + a1 κ 2) ⎡ ω + aj κ ⎤ ⎫ i (−κx + ωt + θ) ψ (x , t ) = csch2 ⎢ ( k ( x − ρt ) + ϑ ) e , 0 ⎥ ⎬ ⎨ κ (β1 + γ1) aj k 2 ⎣ ⎦⎭ ⎩
(66)
1 2 ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) ⎫ i (−κx + ωt + θ) ϕ (x , t ) = csch2 ⎡ (k (x − ρt ) + ϑ0) ⎤ e , 2 ⎢ ⎥ ⎨ κ (β2 + γ2) ⎬ a k 2 ⎣ ⎦⎭ ⎩
where Eqs. (66) and (67) means singular soliton solutions whenever 431
(67)
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
aj (ω + aj κ 2) > 0. 1 2 ω + aj κ 2 ⎧ 2(ω + a1 κ 2) 2 ⎡ ⎤ ⎫ i (−κx + ωt + θ) ψ (x , t ) = − sec ⎢ − ( k ( x − ρt ) + ϑ ) e , 0 ⎥ ⎬ ⎨ κ (β1 + γ1) aj k 2 ⎦⎭ ⎣ ⎩
(68)
1 2 ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) 2 ⎡ ⎫ i (−κx + ωt + θ) ϕ (x , t ) = − sec ⎢ − (k (x − ρt ) + ϑ0) ⎤ e , 2 ⎥ ⎨ κ (β2 + γ2) ⎬ a2 k ⎣ ⎦⎭ ⎩
(69)
1 2
ω + aj κ 2 ⎧ 2(ω + a1 κ 2) 2 ⎡ ⎤⎫ ψ (x , t ) = − csc ⎢ − (k (x − ρt ) + ϑ0) ⎥ ei (−κx + ωt + θ), ⎬ ⎨ κ (β1 + γ1) aj k 2 ⎦⎭ ⎣ ⎩
(70)
1 2
ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) 2 ⎡ ⎫ i (−κx + ωt + θ) ϕ (x , t ) = − csc ⎢ − (k (x − ρt ) + ϑ0) ⎤ , ⎥⎬ e ⎨ κ (β2 + γ2) a2 k 2 ⎣ ⎦⎭ ⎩
(71)
where Eqs. (68) and (71) mean singular periodic solutions whenever
aj (ω + aj κ 2) < 0. 3. Conclusions The coupled system corresponding to Gerdjikov–Ivanov equation, which has an important place to govern parallel transmission of pulses through optical fibers, is addressed by virtue of reaching optical soliton solutions of its in this work. Under the condition of parameter restrictions, bright, dark in addition to singular solitons are catched out through the medium of a procedure that is worked and also is referred to modified simple equation technique. By means of handling the reverse form of the restrictions, additional solutions namely singular periodic solutions have been located. The consequences of this study are encouraging with purpose taking into account the model in more detail. The GI equation with FWM in birefringent fibers will be revisited and moreover is going to be studied through modified simple equation algorithm. Also, the GI equation will be extended in DWDM system. Moreover, additional perturbation terms which are referred to as Raman scattering, multi-photon absorption, saturable amplifiers are going to be involved in future. These quite important consequences are going to be offered by a lot of publications sequentially. References [1] A. Biswas, Y. Yıldırım, E. Yaşar, M.M. Babatin, Conservation laws for Gerdjikov–Ivanov equation in nonlinear fiber optics and PCF, Optik – Int. J. Light Electron Opt. 148 (2017) 209–214. [2] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with Gerdjikov–Ivanov equation by modified simple equation method, Optik – Int. J. Light Electron Opt. 157 (2018) 1235–1240. [3] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Gerdjikov–Ivanov equation by trial equation method, Optik – Int. J. Light Electron Opt. 157 (2018) 1214–1218. [4] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Solitons for perturbed Gerdjikov–Ivanov equation in optical fibers and PCF by extended Kudryashov's method, Opt. Quantum Electron. 50 (3) (2018) 149. [5] H. Triki, R.T. Alqahtani, Q. Zhou, A. Biswas, New envelope solitons for Gerdjikov–Ivanov model in nonlinear fiber optics, Superlattices Microstruct. 111 (2017) 326–334. [6] A. Biswas, R.T. Alqahtani, Chirp-free bright optical solitons for perturbed Gerdjikov–Ivanov equation by semi-inverse variational principle, Optik – Int. J. Light Electron Opt. 147 (2017) 72–76. [7] A. Biswas, M. Ekici, A. Sonmezoglu, F.B. Majid, H. Triki, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for Gerdjikov–Ivanov equation by extended trial equation method, Optik – Int. J. Light Electron Opt. 158 (2018) 747–752. [8] S. Arshed, A. Biswas, M. Abdelaty, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for Gerdjikov–Ivanov equation via two analytical techniques, Chin. J. Phys. 56 (6) (2018) 2879–2886. [9] A. Biswas, M. Ekici, A. Sonmezoglu, H. Triki, A.S. Alshomrani, Q. Zhou, S.P. Moshokoa, M. Belic, Optical solitons for Gerdjikov–Ivanov model by extended trial equation scheme, Optik – Int. J. Light Electron Opt. 157 (2018) 1241–1248. [10] A.J.A.M. Jawad, A. Biswas, M. Abdelaty, Q. Zhou, S.P. Moshokoa, M. Belic, Chirped singular and combo optical solitons for Gerdjikov–Ivanov equation using three integration forms, Optik 172 (2018) 144–149. [11] Y. Yildirim, Optical solitons to Gerdjikov–Ivanov equation in birefringent fibers with trial equation integration architecture, Optik (2019) (in press). [12] Y. Yildirim, Optical solitons of Gerdjikov–Ivanov equation with four-wave mixing terms in birefringent fibers using trial equation scheme, Optik (2019) (in press). [13] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical solitons for Lakshmanan–Porsezian–Daniel model by modified simple equation method, Optik 160 (2018) 24–32. [14] E. Yaşar, Y. Yıldırım, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, H. Triki, A. Biswas, M. Belic, Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method, Superlattices Microstruct. 111 (2017) 487–498. [15] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Kundu–Eckhaus equation by modified simple equation method, Optik – Int. J. Light Electron Opt. 157 (2018) 1376–1380. [16] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with resonant nonlinear Schrödinger's equation having full nonlinearity by modified simple equation method, Optik 160 (2018) 33–43. [17] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for complex Ginzburg–Landau equation with modified simple equation method, Optik – Int. J. Light Electron Opt. 158 (2018) 399–415. [18] Y. Yildirim, Bright, dark and singular optical solitons to Kundu–Eckhaus equation having four-wave mixing in the context of birefringent fibers by using of modified simple equation methodology, Optik (2019). [19] Y. Yildirim, Optical solitons to Schrödinger–Hirota equation in DWDM system with modified simple equation integration architecture, Optik (2019) (in press).
432
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons of Gerdjikov–Ivanov equation in birefringent fibers with modified simple equation scheme
T
Yakup Yildirim Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: Birefringent fibers The coupled Gerdjikov–Ivanov model Modified simple equation algorithm Optical solitons
The coupled system corresponding to Gerdjikov–Ivanov equation is employed to obtain optical solitons of its in this current study. The modified simple equation integration procedure is used to get not only such strategic solitons namely dark, bright with singular solitons but also singular periodic solutions.
1. Introduction Solitons in optics create the basic fabric for soliton transmission technology along with transcontinental and transoceanic distances, optical fibers, data transmission across, telecommunications industry that are constituted through the medium of a lot of models [1–19]. Gerdjikov–Ivanov (GI) model, which has achieved popularity especially in not only photonic crystal fibers but also nonlinear fiber optics since its first appearance, is one of these governing models and furthermore generally has been studied in polarization preserving fibers along with strategic algorithms that are the coupled amplitude-phase methodology, modified simple equation technique, trial equation technique, extended trial equation procedure, extended Kudryashov's method, the extended tanh–coth procedure, semi-inverse variational methodology, the csch scheme, the sine–cosine approach and also G′/G2-expansion technique [1–10]. In despite of these advancements, the solitons along the only one component of the model have been taken into consideration. In order to improve further the model by making contributions to all these studies, the model has been given in two component form for vector solitons with and without four-wave mixing terms (FWM) [11,12] which give rise to improve the model further along. Because of all these reasons, the GI equation without FWM in birefringent fibers, which has an important place to govern parallel transmission of pulses through optical fibers, will be revisited and is going to be investigated along with modified simple equation methodology which recovers strategic solitons namely dark, bright with singular solitons with the inclusion of the existence of circumstances for these important solitons. Additionally, singular-periodic solutions are going to be emerged as a result of the reverse form of the restrictions. In the advancing sections of this work, the details of the analysis of the GI model are given. 1.1. Governing model The Gerdjikov–Ivanov equation [1–10] is imparted by
iψt + aψxx + b|ψ|4 ψ + icψ2ψx* = 0.
(1)
The first term is referred to as the temporal evolution of pulses when the existence of group velocity dispersion is supplied by the coefficient of a in this quite important governing model. The complex valued function ψ(x, t) is referred to as the wave profile. The
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.01.047 Received 19 January 2019; Accepted 19 January 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
coefficient of b is named as the nonlinear term that signifies quintic nonlinearity. Once and for all, the existence of a form of dispersive phenomenon is ensured with the coefficient of c. The Gerdjikov–Ivanov model without FWM in birefringent fibers [11] is described by
iψt + a1 ψxx + (ξ1 |ψ|4 + η1 |ψ|2 |ϕ|2 + ζ1 |ϕ|4 ) ψ + i (β1 ψ2 + γ1 ϕ2) ψx* = 0, iϕt + a2 ϕxx + (ξ2 |ϕ|4 + η2 |ϕ|2 |ψ|2 + ζ2 |ψ|4 ) ϕ + i (β2 ϕ2 + γ2 ψ2) ϕx* = 0.
(2)
The coefficients of ai correspond to group velocity dispersion when the coefficients of ξi stem from self-phase modulation in this coupled GI system. Once and for all, the coefficients of ηi as well as ζi correspond to cross-phase modulation whilst the coefficients of βi, γi account for other forms of dispersive phenomenon along with i = 1, 2. 2. Pen-picture of modified simple equation procedure Succinct overview of modified simple equation algorithm [13–19] is going to be imparted before its feasibility to the coupled GI system in order to locate strategic optical solitons. Step-1: First of all, we will employ a NLEE in general form
Ψ(ζ , ζt , ζ x , ζtt, ζ xt, ζ xx , …) = 0
(3)
and later on inserting the wave variable transformation which is a very important place in this method
ζ (x , t ) = Θ(ϑ)
(4)
along with (5)
ϑ = x − vt into Eq. (3) give rise to
Φ(Θ, Θ′, Θ′ ′, Θ′ ′′, …) = 0
(6)
which is the reduced equation and also known as ordinary differential equation. In (6), the function Θ = Θ(ξ) is referred to as an dependent function and furthermore Φ corresponds to a polynomial with the involving of the dependent function Θ and its derivatives. Step-2: The auxiliary equation that is another key point of this method is going to be addressed by means of N
Θ(ϑ) =
i
Q′ (ϑ)
∑ δi ⎛ Q (ϑ) ⎞ i=0
⎜
⎟
⎝
⎠
(7)
which involves the required coefficients δ0, δ1, …, δN and this coefficients have to be evaluated to get optical solitons. Step-3: The existence of the N number as a result of the application of the balancing principle is vital in the use of the auxiliary equation. Step-4: An overdeterminet equation system are occurred under the condition of inserting Eq. (7) into Eq. (6) and later on putting together the coefficient of the same functions Q−i(ϑ), i = 1, 2, … to zero. As a result of the solution of the system, the required coefficients and Q(ϑ) are granted. 2.1. Soliton solutions The modified simple equation technique, that will be employed on account of locating exact solutions to Eq. (2), is known as one of the most strategic algorithms in literature. In order to get important optical solitons of the coupled GI system, the following solution condition
ψ (x , t ) = P1 (ϑ) eiφ (x , t ), ϕ (x , t ) = P2 (ϑ) eiφ (x , t )
(8)
is going to be taken into consideration. With the inclusion of the parameter
ϑ = k (x − ρt )
(9)
which includes the speed of the soliton ρ, the function Pj(ϑ) accounts for the amplitude component while the function φ(x, t) corresponds to the phase component and also this function will be given as
φ (x , t ) = −κx + ωt + θ .
(10)
In Eq. (10), the parameter κ is referred to as soliton frequency while the parameter ω corresponds to soliton wave number and lastly the parameter θ implies to soliton phase. The real component and the imaginary component are granted by means of
aj k 2P′j ′ − (ω + aj κ 2) Pj − κβj P 3j − κγj Pj P j2˜ + ξ j P 5j + ηj P 3j P j2˜ + ζ j Pj P j˜4 = 0 425
(11)
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
and
− ρ − 2aj κ + βj P j2 + γj P j2˜ = 0
(12)
sequentially through the medium of inserting Eq. (8) into Eq. (2) for j = 1, 2 and j˜ = 3 − j . The following condition easily can be get
P j˜ = Pj
(13)
on account of implement the balancing rule in (11) and this condition convert the real part (11) and the imaginary part (12) into
aj k 2P′j ′ − (ω + aj κ 2) Pj − κ (βj + γj ) P 3j + (ξ j + ηj + ζ j ) P 5j = 0,
(14)
ρ = −2aj κ + (βj + γj ) P j2
(15)
respectively. The speed of the soliton is yield along with Eq. (15) while the profile of the soliton is emerged under the condition of integrating Eq. (14). Case-1: Comparing P′j ′ between P 3j in Eq. (14), one can attain N = 1 so the auxiliary equation (7) is imparted by
Q′ (ϑ) ⎞ Pj (ϑ) = δ0 + δ1 ⎛ . ⎝ Q (ϑ) ⎠ ⎜
⎟
(16)
The following overdeterminet equations can be imparted by means of Q−5 coeff.:
δ15 (Q′)5 (ξ j + ηj + ζ j ) = 0,
(17)
Q−4 coeff.:
5δ0 δ14 (Q′) 4 (ξ j + ηj + ζ j ) = 0, Q
−3
(18)
coeff.:
δ1 (Q′)3 (10δ02 δ12 ηj + 10δ02 δ12 ξ j + 10δ02 δ12 ζ j − κβj δ12 − κδ12 γj + 2k 2aj ) = 0, Q
−2
(19)
coeff.:
− δ1 Q′ ((−10δ03 δ1 ηj − 10δ03 δ1 ξ j − 10δ03 δ1 ζ j + 3κβj δ0 δ1 + 3κδ0 δ1 γj ) Q′ + 3aj k 2Q′ ′) = 0, Q
−1
(20)
coeff.:
− δ1 ((−5δ04 ηj − 5δ04 ξ j − 5δ04 ζ j + 3κβj δ02 + 3κδ02 γj + κ 2aj + ω) Q′ − k 2aj Q′ ′′) = 0,
(21)
Q0 coeff.:
δ0 (δ04 ηj + δ04 ξ j + δ04 ζ j − κβj δ02 − κγj δ02 − κ 2aj − ω) = 0,
(22)
under the condition of inserting Eq. (16) into Eq. (14) and later on putting together the coefficient of the same functions to zero. As a result of the solution of the system (17)–(22), the required coefficients are granted as follows
δ0 = ± −
ω + aj κ 2 κ (βj + γj )
,
δ1 = ±
2aj k 2 κ (βj + γj )
,
ηj = −ξ j − ζ j
and
Q′ ′ = ± −
Q′ ′′ = −
2(ω + aj κ 2)
Q′,
aj k 2
(23)
2(ω + aj κ 2) aj k 2
Q′.
(24)
One can easily reach out to
Q′ = ± −
aj k 2 2(ω + aj
± −
κ 2)
k1 e
2(ω + aj κ 2) aj k 2
ϑ
, (25)
and 426
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
Q=−
aj k 2 2(ω + aj
± −
κ 2)
k1 e
2(ω + aj κ 2)
ϑ
aj k 2
+ k2,
(26)
by means of using Eqs. (23) and (24). On account of inserting Eqs. (25) and (26) which involve k1 and k2 integration constants into Eq. (16), the exact solutions to Eq. (2) are granted by means of
ψ (x , t ) =⎧± ⎨ ⎩
ω + a1 κ 2 1 + γ1)
− κ (β
±
2a1 k2 κ (β1 + γ1)
2
2(ω + a1 κ ) ⎛ k (x − ρt ) ⎞ ⎫ ± − a1 k2 a1 k 2 ⎜ ± − 2(ω + a1κ2) k1 e ⎟ ⎪ i (−κx + ωt + θ) ×⎜ ⎟⎬e 2(ω + a1 κ 2) k (x − ρt ) ± − ⎜ − a1k2 k1 e ⎟⎪ a1 k 2 + k 2 2 ⎝ 2(ω + a1κ ) ⎠⎭
and
ϕ (x , t ) =⎧± ⎨ ⎩
ω + a2 κ 2 2 + γ2)
− κ (β
±
2a2 k2 κ (β2 + γ2)
2
2(ω + a2 κ ) ⎛ k (x − ρt ) ⎞ ⎫ ± − a2 κ 2 a2 k 2 ± − k e 1 ⎟ ⎪ i (−κx + ωt + θ) ⎜ 2(ω + a2 κ 2) ×⎜ ⎟⎬e 2(ω + a2 κ 2) k (x − ρt ) ± − ⎜ − a2 κ2 k1 e a2 k 2 + k2 ⎟ ⎪ 2 ⎠⎭ ⎝ 2(ω + a2 κ )
If we set
k1 = −
2(ω + aj κ 2) aj κ 2
± −
e
2(ω + aj κ 2) aj k 2
ϑ0
,
k2 = ± 1,
we get:
ψ (x , t ) = ± −
ϕ (x , t ) = ± −
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ ) , tanh ⎡ (k (x − ρt ) + ϑ0) ⎤ − 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(27)
ω + a2 κ 2 ω + a2 κ 2 ⎤ i (−κx + ωt + θ), tanh ⎡ ⎢ − 2a2 k 2 (k (x − ρt ) + ϑ0) ⎥ e κ (β2 + γ2) ⎣ ⎦
(28)
where Eqs. (27) and (28) means dark soliton solutions whenever
aj (ω + aj κ 2) < 0. ψ (x , t ) = ± −
ϕ (x , t ) = ± −
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ) , coth ⎡ (k (x − ρt ) + ϑ0) ⎤ − 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(29)
ω + a2 κ 2 ω + a2 κ 2 ⎤ i (−κx + ωt + θ). coth ⎡ ⎢ − 2a2 k 2 (k (x − ρt ) + ϑ0) ⎥ e κ (β2 + γ2) ⎣ ⎦
(30)
These solutions stem from singular soliton solutions under the condition of
aj (ω + aj κ 2) < 0. ψ (x , t ) = ±
ϕ (x , t ) = ±
ψ (x , t ) = ±
ϕ (x , t ) = ±
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ ) , tan ⎡ (k (x − ρt ) + ϑ0) ⎤ 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(31)
ω + a2 κ 2 ω + a2 κ 2 i (−κx + ωt + θ) , tan ⎡ (k (x − ρt ) + ϑ0) ⎤ ⎢ ⎥e κ (β2 + γ2) 2a2 k 2 ⎣ ⎦
(32)
ω + a1 κ 2 ω + a1 κ 2 i (−κx + ωt + θ ) , cot ⎡ (k (x − ρt ) + ϑ0) ⎤ 2 ⎢ ⎥e κ (β1 + γ1) 2 a k 1 ⎣ ⎦
(33)
ω + a2 κ 2 ω + a2 κ 2 i (−κx + ωt + θ) . cot ⎡ (k (x − ρt ) + ϑ0) ⎤ ⎢ ⎥e κ (β2 + γ2) 2a2 k 2 ⎣ ⎦
(34)
427
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
These solutions correspond to singular periodic solutions as long as
aj (ω + aj κ 2) > 0. Case-2: Eq. (14) can easily be turned into the following form
aj k 2 (−(V ′j )2 + 2Vj V ′j ′) − 4(ω + aj κ 2) V j2 − 4κ (βj + γj ) V j3 + 4(ξ j + ηj + ζ j ) V j4 = 0, by means of getting Pj = imparted by
1 V j2 .
Comparing (V ′j )2 or Vj V ′j ′ between V j4 in Eq. (35), one ensure N = 1 so the auxiliary equation (7) is
Q′ (ϑ) ⎞ Vj (ϑ) = δ0 + δ1 ⎛ . ⎝ Q (ϑ) ⎠ ⎜
(35)
⎟
(36)
The following overdeterminet equations can be imparted by means of Q−4 coeff.:
(Q′) 4δ12 (3k 2aj + 4δ12 ηj + 4δ12 ξ j + 4δ12 ζ j ) = 0, Q
−3
(37)
coeff.:
4δ1 (Q′)2 ((k 2aj δ0 − κβj δ12 − κδ12 γj + 4δ0 δ12 ηj + 4δ0 δ12 ξ j + 4δ0 δ12 ζ j ) Q′ − k 2aj δ1 Q′ ′) = 0,
(38)
Q−2 coeff.:
δ1 ((−4κ 2aj δ1 − 12κβj δ0 δ1 − 12κδ0 δ1 γj + 24δ02 δ1 ηj + 24δ02 δ1 ξ j + 24δ02 δ1 ζ j − 4ωδ1)(Q′)2 − 6k 2aj δ0 Q′Q′ ′ + 2k 2aj δ1 Q′Q′ ′′ − k 2aj δ1 (Q′ ′)2) = 0, Q
−1
(39)
coeff.:
2δ0 δ1 ((−4κ 2aj − 6κβj δ0 − 6κδ0 γj + 8δ02 ηj + 8δ02 ξ j + 8δ02 ζ j − 4ω) Q′ + 3k 2aj Q′ ′′) = 0,
(40)
0
Q coeff.:
4δ02 (−κ 2aj − κβj δ0 − κδ0 γj + δ02 ηj + δ02 ξ j + δ02 ζ j − ω) = 0,
(41)
under the condition of inserting Eq. (36) into Eq. (35) and later on putting together the coefficient of the same functions to zero. As a result of the solution of the system (37)–(41), the required coefficients are granted as follows
δ0 =
3κ (βj + γj ) 4(ξ j + ηj + ζ j )
ω=−
,
δ1 = ± −
3aj k 2 4(ξ j + ηj + ζ j )
,
κ 2 (16aj ηj + 16aj ξ j + 16aj ζ j + 3βj2 + 6βj γj + 3γ j2) 16(ξ j + ηj + ζ j )
and
Q′ ′ = ± −
Q′ ′′ = −
3κ 2 (βj + γj )2
Q′,
4aj k 2 (ξ j + ηj + ζ j ) 3κ 2 (βj + γj )2
4aj k 2 (ξ j + ηj + ζ j )
(42)
Q′. (43)
One can easily reach out to
Q′ = ± −
4aj k 2 (ξ j + ηj + ζ j ) 3κ 2 (βj
+ γj
)2
± −
k1 e
3κ 2 (βj + γj )2 4aj k 2 (ξ j + ηj + ζ j )
ϑ
, (44)
and
Q=−
4aj k 2 (ξ j + ηj + ζ j ) 3κ 2 (βj
+ γj
)2
± −
k1 e
3κ 2 (βj + γj )2 4aj k 2 (ξ j + ηj + ζ j )
ϑ
+ k2
(45)
by means of using Eqs. (42) and (43). On account of inserting Eqs.(44) and (45) which involve k1 and k2 integration constants into Eq. 428
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
(36), the exact solutions to Eq. (2) are granted by means of 3κ (β + γ )
ψ (x , t ) =⎧ 4(ξ +1η +1ζ ) ± ⎨ ⎩ 1 1 1
3a1 k2
− 4(ξ
1 + η1 + ζ1)
2
2
3κ (β1+ γ1) ⎛ ± − k (x − ρt ) ⎞ ⎫ 4a1 k2 (ξ1 + η1 + ζ1) 4a1 k 2 (ξ1+ η1+ ζ1) ± − k e 1 ⎟⎪ ⎜ ⎪ i (−κx + ωt + θ) 3κ 2 (β1 + γ1)2 ×⎜ ⎟⎬e 2 2 3κ (β1+ γ1) ⎟⎪ ⎜ 4a1k2 (ξ1 + η1 + ζ1) ± − 4a1k 2 (ξ + η + ζ ) k (x − ρt ) 1 1 1 k1 e + k2 ⎟ ⎜− 2 2 ⎪ ⎠⎭ ⎝ 3κ (β1 + γ1)
and 3κ (β + γ )
ϕ (x , t ) =⎧ 4(ξ +2η +2ζ ) ± ⎨ ⎩ 2 2 2
− 4(ξ
3a2 k2
2 + η2 + ζ2)
2
2
3κ (β2 + γ2) ⎛ ± − k (x − ρt ) ⎞ ⎫ 4a2 k2 (ξ2 + η2 + ζ2) 4a2 k 2 (ξ2+ η2 + ζ2) ± − k e 1 ⎟⎪ ⎜ ⎪ i (−κx + ωt + θ) 3κ 2 (β2 + γ2)2 ×⎜ ⎟⎬e 2 2 3κ (β2 + γ2) ⎟⎪ ⎜ 4a2 k2 (ξ2 + η2 + ζ2) ± − 4a2 k 2 (ξ + η + ζ ) k (x − ρt ) 2 2 2 k1 e + k2 ⎟ ⎜− 2 2 ⎪ ⎠⎭ ⎝ 3κ (β2 + γ2)
If we set
k1 = −
3κ 2 (βj + γj )2 4aj
k 2 (ξ
± −
+ ηj + ζ j )
j
e
3κ 2 (βj + γj )2 4aj k 2 (ξ j + ηj + ζ j )
ϑ0
,
k2 = ± 1,
we get: 3κ (β + γ ) 3κ 2 (β1 + γ1)2 ψ (x , t ) = ⎧ 8(ξ +1η +1ζ ) ⎜⎛1 ± tanh ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎟⎞ ⎫ 16aj k2 (ξ1 + η1 + ζ1) ⎥ ⎢ ⎨ 1 1 1 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
1 2
κ 2 (16a1 η1+ 16a1 ξ1+ 16a1 ζ1+ 3β12 + 6β1 γ1+ 3γ12) ⎛ ⎞ i ⎜−κx − t + θ⎟ 16(ξ1+ η1+ ζ1) ⎝ ⎠,
(46)
3κ (β + γ ) 3κ 2 (β2 + γ2)2 ϕ (x , t ) = ⎧ 8(ξ +2η +2ζ ) ⎛⎜1 ± tanh ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎟⎞ ⎫ 16aj k2 (ξ2 + η2 + ζ2) ⎥ ⎢ ⎨ 2 2 2 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
⎛ i ⎜−κx − ⎝
1 2
κ 2 (16a2 η2 + 16a2 ξ2+ 16a2 ζ2+ 3β22 + 6β2 γ2+ 3γ22) ⎞ t + θ⎟ 16(ξ2+ η2 + ζ2)
(47)
⎠.
These solutions stem from dark soliton solutions under the condition of
aj (ξ j + ηj + ζ j ) < 0. 3κ (β + γ ) 3κ 2 (β1 + γ1)2 ψ (x , t ) = ⎧ 8(ξ +1η +1ζ ) ⎜⎛1 ± coth ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎟⎞ ⎫ 16aj k2 (ξ1 + η1 + ζ1) ⎥ ⎢ ⎨ 1 1 1 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
⎛ i ⎜−κx − ⎝
1 2
κ 2 (16a1 η1+ 16a1 ξ1+ 16a1 ζ1+ 3β12 + 6β1 γ1+ 3γ12) ⎞ t + θ⎟ 16(ξ1+ η1+ ζ1)
⎠,
(48)
3κ (β + γ ) 3κ 2 (β2 + γ2)2 ϕ (x , t ) = ⎧ 8(ξ +2η +2ζ ) ⎛⎜1 ± coth ⎡ − (k (x − ρt ) + ϑ0) ⎤ ⎞⎟ ⎫ 16aj k2 (ξ2 + η2 + ζ2) ⎥ ⎢ ⎨ 2 2 2 ⎝ ⎦⎠⎬ ⎣ ⎭ ⎩
×e
⎛ i ⎜−κx − ⎝
1 2
κ 2 (16a2 η2 + 16a2 ξ2+ 16a2 ζ2+ 3β22 + 6β2 γ2+ 3γ22) ⎞ t + θ⎟ 16(ξ2+ η2 + ζ2)
⎠,
(49)
where Eqs. (48) and (49) singular soliton solutions whenever
aj (ξ j + ηj + ζ j ) < 0. Case-3: Comparing (V ′j )2 or Vj V ′j ′ between V 3j in Eq. (35), one ensure N = 2 so the auxiliary equation (7) is imparted by 2
Q′ (ϑ) ⎞ Q′ (ϑ) ⎞ Vj (ϑ) = δ0 + δ1 ⎛ + δ2 ⎛ . Q (ϑ) ⎝ ⎠ ⎝ Q (ϑ) ⎠ ⎜
⎟
⎜
⎟
(50)
The following overdeterminet equations can be imparted by means of 429
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
Q−8 coeff.:
4δ24 (Q′)8 (ηj + ξ j + ζ j ) = 0, Q
−7
(51)
coeff.:
16δ1 δ23 (Q′)7 (ηj + ξ j + ζ j ) = 0, Q
−6
(52)
coeff.:
4δ22 (Q′)6 (2k 2aj − κβj δ2 − κδ2 γj + 4δ0 δ2 ηj + 4δ0 δ2 ξ j + 4δ0 δ2 ζ j + 6δ12 ηj + 6δ12 ξ j + 6δ12 ζ j ) = 0, Q
−5
(53)
coeff.:
− 4δ2 (Q′) 4 ((−3k 2aj δ1 + 3κβj δ1 δ2 + 3κδ1 δ2 γj − 12δ0 δ1 δ2 ηj − 12δ0 δ1 δ2 ξ j − 12δ0 δ1 δ2 ζ j − 4δ13 ηj − 4δ13 ξ j − 4δ13 ζ j ) Q′ + 3k 2aj δ2 Q′ ′) = 0, Q
−4
(54)
coeff.:
− (Q′)3 ((−3k 2aj δ12 + 4δ22 aj κ 2 − 24ξ j δ02 δ22 − 24ηj δ02 δ22 − 24ζ j δ02 δ22 + 4δ22 ω − 4ξ j δ14 − 4ηj δ14 − 4ζ j δ14 − 12k 2aj δ0 δ2 + 12κβj δ0 δ22 + 12κβj δ12 δ2 + 12κγj δ0 δ22 + 12κγj δ12 δ2 − 48ξ j δ0 δ12 δ2 − 48ηj δ0 δ12 δ2 − 48ζ j δ0 δ12 δ2) Q′ + 18k 2aj δ1 δ2 Q′ ′ − 4k 2aj δ22 Q′ ′′) = 0, Q
−3
(55)
coeff.:
− 2(Q′)2 ((−2k 2aj δ0 δ1 + 4κ 2aj δ1 δ2 + 12κβj δ0 δ1 δ2 + 2κβj δ13 + 12κδ0 δ1 δ2 γj + 2κδ13 γj − 24δ02 δ1 δ2 ηj − 24δ02 δ1 δ2 ξ j − 24δ02 δ1 δ2 ζ j − 8δ0 δ13 ηj − 8δ0 δ13 ξ j − 8δ0 δ13 ζ j + 4ωδ1 δ2) Q′ + (10k 2aj δ0 δ2 + 2k 2aj δ12) Q′ ′ − 3k 2aj δ1 δ2 Q′ ′′) = 0,
(56)
Q−2 coeff.:
(−8κ 2aj δ0 δ2 − 4κ 2aj δ12 − 12κβj δ02 δ2 − 12κβj δ0 δ12 − 12κδ02 δ2 γj − 12κδ0 δ12 γj + 16δ03 δ2 ηj + 16δ03 δ2 ξ j + 16δ03 δ2 ζ j + 24δ02 δ12 ηj + 24δ02 δ12 ξ j + 24δ02 δ12 ζ j − 8ωδ0 δ2 − 4ωδ12)(Q′)2 − 6k 2aj δ0 δ1 Q′Q′ ′ + (4k 2aj δ0 δ2 + 2k 2aj δ12) Q′Q′ ′′ + (4k 2aj δ0 δ2 − k 2aj δ12)(Q′ ′)2 = 0, Q
−1
(57)
coeff.:
− 2δ0 δ1 ((4κ 2aj + 6κβj δ0 + 6κδ0 γj − 8δ02 ηj − 8δ02 ξ j − 8δ02 ζ j + 4ω) Q′ − k 2aj Q′ ′′) = 0,
(58)
0
Q coeff.:
− 4δ02 (κ 2aj + κβj δ0 + κδ0 γj − δ02 ηj − δ02 ξ j − δ02 ζ j + ω) = 0,
(59)
under the condition of inserting Eq. (50) into Eq. (35) and later on putting together the coefficient of the same functions to zero. As a result of the solution of the system (51)–(59), the required coefficients are granted as follows
δ0 = 0,
δ1 = ±
16k 2aj (ω + aj κ 2) κ 2 (βj + γj )2
,
δ2 =
2k 2aj κ (βj + γj )
,
ηj = −ξ j − ζ j
and
Q′ ′ = ±
Q′ ′′ =
4(ω + aj κ 2) aj k 2
4(ω + aj aj k 2
κ 2)
Q′, (60)
Q′.
(61)
One can easily reach out to 430
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
Q′ = ±
aj k 2
±
4(ω + aj
4(ω + aj κ 2)
k1 e
κ 2)
aj k 2
ϑ
, (62)
and
Q=
aj k 2
±
4(ω + aj κ 2)
k1 e
4(ω + aj κ 2)
ϑ
aj k 2
+ k2
(63)
by means of using Eqs. (60) and (61). On account of inserting Eqs. (62) and (63) which involve k1 and k2 integration constants into Eq. (50), the exact solutions to Eq. (2) are granted by means of 2
⎧ ⎪ ψ (x , t ) =
⎨ ⎪ ⎩
±
16k2a1 (ω + a1 κ 2) κ 2 (β1 + γ1)2
4(ω + a1 κ ) ⎛ k (x − ρt ) ⎞ ± a1 k2 a1 k 2 ⎟ ⎜ ± 4(ω + a1κ2) k1 e ⎟ ⎜ 4(ω + a1 κ 2) k (x − ρt ) ± ⎟ ⎜ a1k2 k1 e a1 k 2 + k 2 2 ⎠ ⎝ 4(ω + a1κ ) 2
1 2 2
⎫ ⎪ ⎪ i (−κx + ωt + θ) e ⎬ ⎪ ⎪ ⎭
4(ω + a1 κ ) ⎛ k (x − ρt ) ⎞ ± a1 k2 a1 k 2 ± k1 e ⎟ 2 ⎜ 2 4(ω + a1 κ ) 2k a1 + κ (β + γ ) ⎜ ⎟ 1 1 4(ω + a1 κ 2) k (x − ρt ) ± ⎜ a1k2 k1 e a1 k 2 + k2 ⎟ 2 ⎠ ⎝ 4(ω + a1κ )
and 2
⎧ ⎪ ϕ (x , t ) =
⎨ ⎪ ⎩
±
16k2a2 (ω + a2 κ 2) κ 2 (β2 + γ2)2
4(ω + a2 κ ) ⎛ k (x − ρt ) ⎞ ± a2 k2 a2 k 2 ⎟ ⎜ ± 4(ω + a2 κ2) k1 e ⎟ ⎜ 4(ω + a2 κ 2) ± k (x − ρt ) ⎜ a2 k2 k1 e a2 k 2 + k2 ⎟ 2 ⎠ ⎝ 4(ω + a2 κ ) 2
1 2 2
4(ω + a2 κ ) ⎛ ± k (x − ρt ) ⎞ a2 k2 a2 k 2 ± k1 e ⎟ ⎜ 2 4(ω + a2 κ ) 2k2a2 + κ (β + γ ) ⎜ ⎟ 2 2 4(ω + a2 κ 2) ± k (x − ρt ) ⎟ ⎜ a2 k2 k1 e a2 k 2 + k 2 2 ⎠ ⎝ 4(ω + a2 κ )
⎫ ⎪ ⎪ i (−κx + ωt + θ) e ⎬ ⎪ ⎪ ⎭
If we set
k1 =
4(ω + aj κ 2) aj
k2
±
e
4(ω + aj κ 2) aj k 2
ϑ0
,
k2 = ± 1,
we get: 1 2 2 ⎧ 2(ω + a1 κ 2) ⎡ ω + aj κ ⎤⎫ ψ (x , t ) = − sech2 ⎢ (k (x − ρt ) + ϑ0) ⎥ ei (−κx + ωt + θ), 2 ⎨ κ (β1 + γ1) ⎬ aj k ⎣ ⎦⎭ ⎩
(64)
1 2 ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) ⎫ i (−κx + ωt + θ) ϕ (x , t ) = − sech2 ⎡ (k (x − ρt ) + ϑ0) ⎤ e , 2 ⎢ ⎥ ⎨ κ (β2 + γ2) ⎬ a2 k ⎣ ⎦⎭ ⎩
(65)
where Eqs. (64) and (65) means bright soliton solutions whenever
aj (ω + aj κ 2) > 0. 1 2 2 ⎧ 2(ω + a1 κ 2) ⎡ ω + aj κ ⎤ ⎫ i (−κx + ωt + θ) ψ (x , t ) = csch2 ⎢ ( k ( x − ρt ) + ϑ ) e , 0 ⎥ ⎬ ⎨ κ (β1 + γ1) aj k 2 ⎣ ⎦⎭ ⎩
(66)
1 2 ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) ⎫ i (−κx + ωt + θ) ϕ (x , t ) = csch2 ⎡ (k (x − ρt ) + ϑ0) ⎤ e , 2 ⎢ ⎥ ⎨ κ (β2 + γ2) ⎬ a k 2 ⎣ ⎦⎭ ⎩
where Eqs. (66) and (67) means singular soliton solutions whenever 431
(67)
Optik - International Journal for Light and Electron Optics 182 (2019) 424–432
Y. Yildirim
aj (ω + aj κ 2) > 0. 1 2 ω + aj κ 2 ⎧ 2(ω + a1 κ 2) 2 ⎡ ⎤ ⎫ i (−κx + ωt + θ) ψ (x , t ) = − sec ⎢ − ( k ( x − ρt ) + ϑ ) e , 0 ⎥ ⎬ ⎨ κ (β1 + γ1) aj k 2 ⎦⎭ ⎣ ⎩
(68)
1 2 ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) 2 ⎡ ⎫ i (−κx + ωt + θ) ϕ (x , t ) = − sec ⎢ − (k (x − ρt ) + ϑ0) ⎤ e , 2 ⎥ ⎨ κ (β2 + γ2) ⎬ a2 k ⎣ ⎦⎭ ⎩
(69)
1 2
ω + aj κ 2 ⎧ 2(ω + a1 κ 2) 2 ⎡ ⎤⎫ ψ (x , t ) = − csc ⎢ − (k (x − ρt ) + ϑ0) ⎥ ei (−κx + ωt + θ), ⎬ ⎨ κ (β1 + γ1) aj k 2 ⎦⎭ ⎣ ⎩
(70)
1 2
ω + a2 κ 2 ⎧ 2(ω + a2 κ 2) 2 ⎡ ⎫ i (−κx + ωt + θ) ϕ (x , t ) = − csc ⎢ − (k (x − ρt ) + ϑ0) ⎤ , ⎥⎬ e ⎨ κ (β2 + γ2) a2 k 2 ⎣ ⎦⎭ ⎩
(71)
where Eqs. (68) and (71) mean singular periodic solutions whenever
aj (ω + aj κ 2) < 0. 3. Conclusions The coupled system corresponding to Gerdjikov–Ivanov equation, which has an important place to govern parallel transmission of pulses through optical fibers, is addressed by virtue of reaching optical soliton solutions of its in this work. Under the condition of parameter restrictions, bright, dark in addition to singular solitons are catched out through the medium of a procedure that is worked and also is referred to modified simple equation technique. By means of handling the reverse form of the restrictions, additional solutions namely singular periodic solutions have been located. The consequences of this study are encouraging with purpose taking into account the model in more detail. The GI equation with FWM in birefringent fibers will be revisited and moreover is going to be studied through modified simple equation algorithm. Also, the GI equation will be extended in DWDM system. Moreover, additional perturbation terms which are referred to as Raman scattering, multi-photon absorption, saturable amplifiers are going to be involved in future. These quite important consequences are going to be offered by a lot of publications sequentially. References [1] A. Biswas, Y. Yıldırım, E. Yaşar, M.M. Babatin, Conservation laws for Gerdjikov–Ivanov equation in nonlinear fiber optics and PCF, Optik – Int. J. Light Electron Opt. 148 (2017) 209–214. [2] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with Gerdjikov–Ivanov equation by modified simple equation method, Optik – Int. J. Light Electron Opt. 157 (2018) 1235–1240. [3] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Gerdjikov–Ivanov equation by trial equation method, Optik – Int. J. Light Electron Opt. 157 (2018) 1214–1218. [4] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Solitons for perturbed Gerdjikov–Ivanov equation in optical fibers and PCF by extended Kudryashov's method, Opt. Quantum Electron. 50 (3) (2018) 149. [5] H. Triki, R.T. Alqahtani, Q. Zhou, A. Biswas, New envelope solitons for Gerdjikov–Ivanov model in nonlinear fiber optics, Superlattices Microstruct. 111 (2017) 326–334. [6] A. Biswas, R.T. Alqahtani, Chirp-free bright optical solitons for perturbed Gerdjikov–Ivanov equation by semi-inverse variational principle, Optik – Int. J. Light Electron Opt. 147 (2017) 72–76. [7] A. Biswas, M. Ekici, A. Sonmezoglu, F.B. Majid, H. Triki, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for Gerdjikov–Ivanov equation by extended trial equation method, Optik – Int. J. Light Electron Opt. 158 (2018) 747–752. [8] S. Arshed, A. Biswas, M. Abdelaty, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for Gerdjikov–Ivanov equation via two analytical techniques, Chin. J. Phys. 56 (6) (2018) 2879–2886. [9] A. Biswas, M. Ekici, A. Sonmezoglu, H. Triki, A.S. Alshomrani, Q. Zhou, S.P. Moshokoa, M. Belic, Optical solitons for Gerdjikov–Ivanov model by extended trial equation scheme, Optik – Int. J. Light Electron Opt. 157 (2018) 1241–1248. [10] A.J.A.M. Jawad, A. Biswas, M. Abdelaty, Q. Zhou, S.P. Moshokoa, M. Belic, Chirped singular and combo optical solitons for Gerdjikov–Ivanov equation using three integration forms, Optik 172 (2018) 144–149. [11] Y. Yildirim, Optical solitons to Gerdjikov–Ivanov equation in birefringent fibers with trial equation integration architecture, Optik (2019) (in press). [12] Y. Yildirim, Optical solitons of Gerdjikov–Ivanov equation with four-wave mixing terms in birefringent fibers using trial equation scheme, Optik (2019) (in press). [13] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical solitons for Lakshmanan–Porsezian–Daniel model by modified simple equation method, Optik 160 (2018) 24–32. [14] E. Yaşar, Y. Yıldırım, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, H. Triki, A. Biswas, M. Belic, Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method, Superlattices Microstruct. 111 (2017) 487–498. [15] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Kundu–Eckhaus equation by modified simple equation method, Optik – Int. J. Light Electron Opt. 157 (2018) 1376–1380. [16] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with resonant nonlinear Schrödinger's equation having full nonlinearity by modified simple equation method, Optik 160 (2018) 33–43. [17] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for complex Ginzburg–Landau equation with modified simple equation method, Optik – Int. J. Light Electron Opt. 158 (2018) 399–415. [18] Y. Yildirim, Bright, dark and singular optical solitons to Kundu–Eckhaus equation having four-wave mixing in the context of birefringent fibers by using of modified simple equation methodology, Optik (2019). [19] Y. Yildirim, Optical solitons to Schrödinger–Hirota equation in DWDM system with modified simple equation integration architecture, Optik (2019) (in press).
432