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Optical solitons to Sasa-Satsuma model in birefringent fibers with modified simple equation approach
Optik - International Journal for Light and Electron Optics 184 (2019) 197–204
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Original research article
Optical solitons to Sasa-Satsuma model in birefringent fibers with modified simple equation approach
T
Yakup Yıldırım 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: Modified simple equation architecture Sasa-Satsuma model Optical soliton pulses
Optical soliton pulses or molecules with the Sasa-Satsuma equation in birefringent fibers have been worked by modified simple equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.
1. Introduction Optical soliton pulses or molecules constitute of the basic fabric for soliton transmission technology such as transcontinental and transoceanic distances, data transmissionacross, optical fibers and telecommunications industry. The dynamics of these molecules or pulses have scuplted the technology to ionospherical level. These developments have prompted more comprehensive researches in this field from an physics and engineering aspect. One of the areas of influential research in this field is the work of the governing model along with fiber nonlinearities, for example, Lakshmanan–Porsezian–Daniel equation, Schrödinger equation with quadraticcubic nonlinearity, Kaup–Newell equation, Gerdjikov–Ivanov model, Schrödinger–Hirota equation, complex Ginzburg–Landau model, Radhakrishnan–Kundu–Lakshmanan equation, Fokas–Lenells equation, Schrödinger equation with weak non-local nonlinearity, Chen–Lee–Liu model, Triki-Biswas model, Biswas-Arshed model, DWDM system and many more [6–31] where soliton pulses along the one, two and N component of the models have been taken into consideration. Moreover, Sasa-Satsuma equation also is given as a model describing this phenomena. This model is an extension of Schrödinger equation which includes the self-steepening, third-order dispersion in additionally stimulated Raman scattering effects in monomode optical fibers. Additionally, this model describes the propagation of femtosecond pulses in optical fibers and also governs the propagation and interaction of the ultrashort pulses in the sub-picosecond or femtosecond regime. Because of this reason, finding optical soliton pulses or molecules to the model in birefringent fibers getting out of four-wave mixing terms (FWM) becomes very important. To achieve this goal, this article investigates the optical soliton pulses or molecules using modified simple equation architecture. In addition to such soliton type solutions, this method give rise to singular-periodic solutions. An important point in obtaining these solutions is parameter restrictions because these parameters compensate these optical soliton pulses or molecules. The detailed steps of obtaining the solutions of the aforementioned governing model are given the following chapters. 1.1. Governing model The Sasa-Satsuma governing model [1–5] can be given as follows
iψt + aψxx + b |ψ|2 ψ + i [αψxxx + β |ψ|2 ψx + θ (|ψ|2 )x ψ] = 0.
(1)
The temporal evolution of optical soliton molecules is given with the first term and also the Kerr law fiber nonlinearity is yield with the coefficient of b. Moreover, the group velocity dispersion term is provided with the coefficient of a and the profile of the optical https://doi.org/10.1016/j.ijleo.2019.03.022 Received 7 March 2019; Accepted 7 March 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 184 (2019) 197–204
Y. Yıldırım
soliton pulses is compensated by ψ(x, t). Lastly, the self-steepening, stimulated Raman scattering in additionally third-order dispersion sequentially are given with the coefficient of β, θ, α. The Sasa-Satsuma model in birefringent fibers without FWM to be discussed in this article is given as follows
iψt + a1 ψxx + (b1 |ψ|2 + c1 |ϕ|2 ) ψ + i [α1 ψxxx + γ1 ϕxxx + (β1 |ψ|2 + η1 |ϕ|2 ) ψx + (θ1 (|ψ|2 )x + λ1 (|ϕ|2 )x ) ψ] = 0, iϕt + a2 ϕxx + (b2 |ϕ|2 + c2 |ψ|2 ) ϕ
+ i [α2 ϕxxx + γ2 ψxxx + (β2 |ϕ|2 + η2 |ψ|2 ) ϕx + (θ2 (|ϕ|2 )x + λ2 (|ψ|2 )x ) ϕ] = 0
(2)
with the inclusion of the self-steepening, stimulated Raman scattering in additionally third-order dispersion that sequentially are given with the coefficient of βj, ηj, θj, λj, αj, γj. 2. A quick glance over modified simple equation technique In this chapter of the manuscript, iteration of the modified simple equation scheme [6–21] is given to work optical soliton molecules. Step-1: A nonlinear evolution equation used in the modeling of any physical event such as soliton transmission technology can be yield as (3)
V (r , rt , rx , rtt, rxt, rxx , …) = 0 and also this strategic equation can be collapsed into the following ordinary differential equation
F (P , P′, P ″, P″ ′, …) = 0
(4)
by using of
r (x , t ) = P (φ)
(5)
as long as
φ = x − ρt.
(6)
Step-2: The auxiliary equation corresponding to the ordinary differential equation can be given N
P (φ) =
Q′ (φ)
i
∑ δi ⎛ Q (φ) ⎞ i=0
⎜
⎟
⎝
⎠
(7)
having the essential constant coefficients δ0, δ1, …, δN where the N number comes from the balancing rule in Eq. (4). Step-3: The overdeterminet equations needed to find optical soliton pulses to Eq. (3) can be yield if Eq. (7) is put in Eq. (4) and subsequently the coefficients of Q−l are supposed as zero sequentially. Thus, the coefficients in the auxiliary equation are given if the overdeterminet equations can be solved. As a result, the optical soliton pulses of Eq. (3) can be yield by using of the obtained coefficients in the auxiliary equation. 2.1. Optical soliton pulses to the model In order to work optical soliton pulses with the Sasa-Satsuma equation, the following transformation can be yield as
ψ (x , t ) = P1 (ϑ) eiφ (x , t ) ϕ (x , t ) = P2 (ϑ) eiφ (x , t )
(8)
ϑ = x − ρt ,
(9)
with
φ (x , t ) = −κx + ωt + ζ .
(10)
The phase and amplitude component sequentially is given with the functions φ(x, t), Pj(ϑ) while the frequency, phase in additionally wave number respectively are given as the parameters κ, ζ, ω. The real and imaginary components sequentially can be yield as
(aj + 3κ αj ) P″j + 3κ γj P ″j˜ − (ω + κ 2aj + κ 3αj ) Pj − κ 3γj P j˜ + (bj + κ βj ) P 3j + (cj + κ ηj ) Pj P j2˜ = 0,
(11)
αj P″j ′ + γj P ″j˜ ′ − (2 κ aj + ρ + 3 κ 2αj ) P′j − 3κ 2γj P ′j˜ + (2 θj + βj ) P′j P j2 + ηj P′j P j2˜ + 2λj Pj P j˜ P ′j˜ = 0
(12)
for j = 1, 2 with j˜ = 3 − j if Eq. (8) is put in Eq. (2). The following condition can be yield as 198
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Y. Yıldırım
P j˜ = Pj
(13)
which give rise to
(aj + 3κ αj + 3κ γj ) P″j − (ω + κ 2aj + κ 3αj + κ 3γj ) Pj + (bj + κ βj + cj + κ ηj ) P 3j = 0, 2 θj + βj + ηj + 2λj
(αj + γj ) P″j − (2 κ aj + ρ + 3 κ 2αj + 3κ 2γj ) Pj +
3
(14)
P 3j = 0
(15)
because of the balancing rule for the imaginary and real parts (11) and (12). We can given the following strategic conditions
aj + 3κ αj + 3κ γj αj + γj
=
ω + κ 2aj + κ 3αj + κ 3γj 2 κ aj + ρ +
3 κ 2α
j
+
3κ 2γj
=
3(bj + κ βj + cj + κ ηj ) 2 θj + βj + ηj + 2λj
(16)
which give rise to
ω=
8 κ 3αj 2 + 16 κ 3αj γj + 8 κ 3γj 2 + 8 κ 2aj αj + 8 κ 2aj γj + 3 κ ρ αj + 3 κ ρ γj + 2 κ aj 2 + ρ aj
βj = −
αj + γj
,
6 κ αj λj + 6 κ αj θj + 6 κ γj λj + 6 κ γj θj + aj ηj + 2 aj λj + 2 aj θj − 3 αj bj − 3 αj cj − 3 bj γj − 3 cj γj aj
(17)
because the amplitude component Pj holds not only Eq. (14) but also Eq. (15). In order to obtain optical soliton molecules to Sasa-Satsuma governing model discussed in this study with modified simple equation scheme, Eq. (14) is considered in the rest of the article along with Eq. (17). Case-1: The auxiliary equation corresponding to the real component can be given
Q′ (ϑ) ⎞ Pj (ϑ) = δ0 + δ1 ⎛ ⎝ Q (ϑ) ⎠ ⎜
⎟
(18)
P 3j .
which comes from the balancing rule between the terms P″j and The overdetermined equations needed to find optical soliton pulses to Sasa-Satsuma governing model discussed in this paper can be yield as Q−3 coeff.:
δ1 (Q′)3 (κ βj δ12 + κ δ12ηj + bj δ12 + cj δ12 + 6 κ αj + 6 κ γj + 2 aj ) = 0, Q
−2
(19)
coeff.:
3δ1 (Q′)((−3 κ αj − 3 κ γj − aj ) Q″ + (κ βj δ0 δ1 + κ δ0 δ1 ηj + bj δ0 δ1 + cj δ0 δ1 ) Q′) = 0, Q
−1
(20)
coeff.:
δ1 ((3 κ αj + 3 κ γj + aj ) Q″ ′ + (−κ 3αj − κ 3γj + 3 κ βj δ0 2 + 3 κ δ0 2ηj
− κ 2aj + 3 bj δ0 2 + 3 cj δ0 2 − ω) Q′) = 0,
(21)
0
Q coeff.:
− κ 3αj δ0 − κ 3δ0 γj + κ βj δ03 + κ δ03ηj − κ 2aj δ0 + bj δ03 + cj δ03 − ω δ0 = 0, if Eq. (18) is put in Eq. (14) and then the coefficients of Q the auxiliary equation are given as
δ0 = ±
κ 3αj + κ 3γj + κ 2aj + ω κ βj + κ ηj + bj + cj
,
δ1 = ± −
−3
,Q
−2
,Q
−1
(22)
0
, Q are supposed as zero sequentially. Thus, the coefficients in
6 κ αj + 6 κ γj + 2 aj κ βj + κ ηj + bj + cj
(23)
and also we get
Q″ = ± −
Q″ ′ = −
2(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
2(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
Q′, (24)
Q′. (25)
If we use Eqs. (24) and (25), we get: 199
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Y. Yıldırım
Q′ = ± −
3 κ αj + 3 κ γj + aj 2(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
± −
2(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
, (26)
and
Q=−
3 κ αj + 3 κ γj + aj 2(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
± −
2(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
+ k2
(27)
with k1 and k2 integration constants. The optical soliton pulses of the Sasa-Satsuma governing model can be yield as follows
κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎧ ψ (x , t ) = ± ± ⎨ κ β1 + κ η1 + b1 + c1 ⎩ 3
3
−
6 κ α1 + 6 κ γ1 + 2 a1 κ β1 + κ η1 + b1 + c1
2
2(κ α1+ κ γ + κ a1+ ω) ⎛ 3 κ α1 + 3 κ γ1 + a1 ± − 3 κ α + 31κ γ + a (x − ρt ) ⎞ ⎫ 1 1 1 ⎟ ⎪ i (−κx + ωt + ζ ) ⎜ ± − 2(κ3α1 + κ3γ1 + κ 2a1 + ω) k1 e ×⎜ , ⎟⎬e 3 3 2 2(κ α1+ κ γ + κ a1+ ω) 3 κ α1 + 3 κ γ1 + a1 ± − 3 κ α + 31κ γ + a (x − ρt ) ⎜− 3 1 1 1 + k2 ⎟ ⎪ k1 e 3 2 ⎠⎭ ⎝ 2(κ α1 + κ γ1 + κ a1 + ω)
κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎧ ϕ (x , t ) = ± ± ⎨ κ β2 + κ η2 + b2 + c2 ⎩ 3
3
−
(28)
6 κ α2 + 6 κ γ2 + 2 a2 κ β2 + κ η2 + b2 + c2
2
2(κ α + κ γ + κ a + ω) ⎛ 3 κ α2 + 3 κ γ2 + a2 ± − 3 κ2α + 32κ γ + a2 (x − ρt ) ⎞ ⎫ 2 2 2 ⎟ ⎪ i (−κx + ωt + ζ ) ⎜ ± − 2(κ3α2 + κ3γ2 + κ 2a2 + ω) k1 e ×⎜ ⎟⎬e 3 3 2 2(κ α + κ γ + κ a + ω) 3 κ α2 + 3 κ γ2 + a2 ± − 3 κ2α + 32κ γ + a2 (x − ρt ) ⎜− 3 2 2 2 + k2 ⎟ ⎪ k1 e 3 2 ⎠⎭ ⎝ 2(κ α2 + κ γ2 + κ a2 + ω)
(29)
and also if we set
k1 = −
2(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
e
± −
2(κ 3αj + κ 3γj + κ 2aj + ω) ϑ0 3 κ αj + 3 κ γj + aj
,
k2 = ± 1,
(30)
in Eqs. (28) and (29), the following strategic dark optical soliton in additionally singular optical soliton are given as
ψ (x , t ) = ±
ϕ (x , t ) = ±
κ 3α1 + κ 3γ1 + κ 2a1 + ω κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎤ ⎡ tanh ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎦ ⎣
(31)
κ 3α2 + κ 3γ2 + κ 2a2 + ω κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎤ ⎡ tanh ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎦ ⎣
(32)
The results (31) and (32) mean dark soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) < 0. ψ (x , t ) = ±
ϕ (x , t ) = ±
κ 3α1 + κ 3γ1 + κ 2a1 + ω κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎤ ⎡ coth ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎦ ⎣
(33)
κ 3α2 + κ 3γ2 + κ 2a2 + ω κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎤ ⎡ coth ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎦ ⎣
(34)
The results (33) and (34) mean singular soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) < 0. ψ (x , t ) = ± −
ϕ (x , t ) = ± −
3 3 2 κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎡ κ α1 + κ γ1 + κ a1 + ω ⎤ tan ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎣ ⎦
(35)
3 3 2 κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎡ κ α2 + κ γ2 + κ a2 + ω ⎤ tan ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎣ ⎦
(36)
200
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ψ (x , t ) = ± −
ϕ (x , t ) = ± −
3 3 2 κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎡ κ α1 + κ γ1 + κ a1 + ω ⎤ cot ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎣ ⎦
(37)
3 3 2 κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎤ ⎡ κ α2 + κ γ2 + κ a2 + ω cot ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎦ ⎣
(38)
The consequences (35)–(38) mean singular periodic solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) > 0. Case-2: We can get the new real component as follows
(aj + 3κ αj + 3κ γj )(−(V ′j )2 + 2Vj V ′′j ) − 4(ω + κ 2aj + κ 3αj + κ 3γj ) V j2
+ 4(bj + κ βj + cj + κ ηj ) V 3j = 0 by using of Pj =
1 V j2
(39)
and moreover the auxiliary equation corresponding to this new equation can be given 2
Q′ (ϑ) ⎞ Q′ (ϑ) ⎞ Vj (ϑ) = δ0 + δ1 ⎛ + δ2 ⎛ Q (ϑ) ⎝ ⎠ ⎝ Q (ϑ) ⎠ ⎜
⎟
⎜
⎟
(40)
(V ′j )2 , V 3j
V 3j ,
or which comes from the balancing rule between the terms either Vj V ″j . The overdeterminet equations needed to find optical soliton pulses to Sasa-Satsuma governing model discussed in this paper can be yield as Q−6 coeff.:
4δ22 (Q′)6 (κ βj δ2 + κ δ2 ηj + 6 κ αj + 6 κ γj + bj δ2 + cj δ2 + 2 aj ) = 0, Q
−5
(41)
coeff.:
12δ2 (Q′) 4 ((−3 κ αj δ2 − 3 κ δ2 γj − aj δ2 ) Q″ + (κ βj δ1 δ2 + κ δ1 δ2 ηj + 3 κ αj δ1 + 3 κ δ1 γj + bj δ1 δ2 + cj δ1 δ2 + aj δ1) Q′) = 0, Q
−4
(42)
coeff.:
− (Q′)3 ((−12 δ22κ αj − 12 δ22κ γj − 4 δ22aj ) Q″ ′ + (54 κ αj δ1 δ2 + 54 κ δ1 δ2 γj + 18 aj δ1 δ2 ) Q″ + (4 κ 3αj δ22 + 4 κ 3δ22γj + 4 κ 2aj δ22 − 12 κ βj δ0 δ22 − 12 κ βj δ12δ2 − 12 κ δ0 δ22ηj
− 12 κ δ12δ2 ηj − 36 κ αj δ0 δ2 − 9 κ αj δ12 − 36 κ δ0 δ2 γj − 9 κ δ12γj − 12 bj δ0 δ22 − 12 bj δ12δ2 − 12 cj δ0 δ22 − 12 cj δ12δ2 + 4 ω δ22 − 12 aj δ0 δ2 − 3 aj δ12) Q′) = 0, Q
−3
(43)
coeff.:
− 2(Q′)2 ((−9 κ αj δ1 δ2 − 9 κ δ1 δ2 γj − 3 aj δ1 δ2 ) Q″ ′ + (30 κ αj δ0 δ2 + 6 κ αj δ12 + 30 κ δ0 δ2 γj + 6 κ δ12γj + 10 aj δ0 δ2 + 2 aj δ12) Q″ + (4 κ 3αj δ1 δ2 + 4 κ 3δ1 δ2 γj + 4 κ 2aj δ1 δ2
− 12 κ βj δ0 δ1 δ2 − 2 κ βj δ13 − 12 κ δ0 δ1 δ2 ηj − 2 κ δ13ηj − 6 κ αj δ0 δ1 − 6 κ δ0 δ1 γj
− 12 bj δ0 δ1 δ2 − 2 bj δ13 − 12 cj δ0 δ1 δ2 − 2 cj δ13 + 4 ω δ1 δ2 − 2 aj δ0 δ1) Q′) = 0, Q
−2
(44)
coeff.:
(12 κ αj δ0 δ2 + 6 κ αj δ12 + 12 κ δ0 δ2 γj + 6 κ δ12γj + 4 aj δ0 δ2 + 2 aj δ12) Q′Q″ ′ + (12 κ αj δ0 δ2 − 3 κ αj δ12 + 12 κ δ0 δ2 γj − 3 κ δ12γj + 4 aj δ0 δ2 − aj δ12)(Q′ ′)2
+ (−18 κ αj δ0 δ1 − 18 κ δ0 δ1 γj − 6 aj δ0 δ1 ) Q′Q″ + (−8 κ 3αj δ0 δ2 − 4 κ 3αj δ12
− 8 κ 3δ0 δ2 γj − 4 κ 3δ12γj − 8 κ 2aj δ0 δ2 − 4 κ 2aj δ12 + 12 κ βj δ0 2δ2 + 12 κ βj δ0 δ12
+ 12 κ δ0 2δ2 ηj + 12 κ δ0 δ12ηj + 12 bj δ0 2δ2 + 12 bj δ0 δ12 + 12 cj δ0 2δ2
+ 12 cj δ0 δ12 − 8 ω δ0 δ2 − 4 ω δ12)(Q′)2 = 0,
(45) 201
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Q−1 coeff.:
− 2δ0 δ1 ((−3 κ αj − 3 κ γj − aj ) Q″ ′ + (4 κ 3αj + 4 κ 3γj + 4 κ 2aj − 6 κ βj δ0
− 6 κ δ0 ηj − 6 bj δ0 − 6 cj δ0 + 4 ω) Q′) = 0,
(46)
Q0 coeff.:
− 4 κ 3αj δ0 2 − 4 κ 3δ0 2γj − 4 κ 2aj δ0 2 + 4 κ βj δ03 + 4 κ δ03ηj + 4 bj δ03 + 4 cj δ03 − 4 ω δ0 2 = 0,
(47) −l
if Eq. (40) is put in Eq. (39) and then the coefficients of Q equation are given as
δ0 = 0,
δ2 = −
2(3 κ αj + 3 κ γj + aj )
are supposed as zero sequentially. Thus, the coefficients in the auxiliary
,
κ βj + κ ηj + bj + cj
16(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω)
δ1 = ±
(κ βj + κ ηj + bj + cj )2
(48)
and also we get
Q″ = ±
Q″ ′ =
4(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
4(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
Q′, (49)
Q′. (50)
If we use Eqs. (49) and (50), we get:
Q′ = ±
3 κ αj + 3 κ γj + aj 4(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
±
4(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
, (51)
and
Q=
3 κ αj + 3 κ γj + aj 4(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
±
4(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
+ k2
(52)
with k1 and k2 integration constants. The optical soliton pulses of the Sasa-Satsuma governing model can be yield as follows
16(3 κ α1 + 3 κ γ1 + a1 )(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎧ ψ (x , t ) = ± ⎨ (κ β1 + κ η1 + b1 + c1 )2 ⎩ 3
3
2
4(κ α1+ κ γ1+ κ a1+ ω) ⎛ ± (x − ρt ) ⎞ 3 κ α1 + 3 κ γ1 + a1 3 κ α1+ 3 κ γ1+ a1 ⎜ ± 4(κ3α1 + κ3γ1 + κ 2a1 + ω) k1 e ⎟ 2(3 κ α1 + 3 κ γ1 + a1 ) ×⎜ ⎟− κβ +κη +b +c 3 3 2 4(κ α1+ κ γ1+ κ a1+ ω) 1 1 1 1 (x − ρt ) ⎜⎜ 3 κ α1 + 3 κ γ1 + a1 k1 e ± 3 κ α1+ 3 κ γ1+ a1 + k2 ⎟⎟ 4(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎝ ⎠
3
3
1 2 2
2
4(κ α1+ κ γ1+ κ a1+ ω) ⎛ ± (x − ρt ) ⎞ 3 κ α1 + 3 κ γ1 + a1 3 κ α1+ 3 κ γ1+ a1 ⎜ ± 4(κ3α1 + κ3γ1 + κ 2a1 + ω) k1 e ⎟ ×⎜ ⎟ 4(κ3α1+ κ3γ1+ κ 2a1+ ω) (x − ρt ) ⎟ ⎜⎜ 3 κ α1 + 3 κ γ1 + a1 k1 e ± 3 κ α1+ 3 κ γ1+ a1 + k 2⎟ 3 3 2 ⎝ 4(κ α1 + κ γ1 + κ a1 + ω) ⎠
⎫ ⎪ ⎪ i (−κx + ωt + ζ ) , e ⎬ ⎪ ⎪ ⎭
16(3 κ α2 + 3 κ γ2 + a2 )(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎧ ϕ (x , t ) = ± ⎨ (κ β2 + κ η2 + b2 + c2 )2 ⎩ 3
3
2
4(κ α2+ κ γ2+ κ a2 + ω) ⎛ ± (x − ρt ) ⎞ 3 κ α2 + 3 κ γ2 + a2 3 κ α2+ 3 κ γ2+ a2 ⎜ ± 4(κ3α2 + κ3γ2 + κ 2a2 + ω) k1 e ⎟ 2(3 κ α2 + 3 κ γ2 + a2 ) ×⎜ ⎟− κβ +κη +b +c 4(κ3α2+ κ3γ2+ κ 2a2 + ω) 2 2 2 2 (x − ρt ) ⎜⎜ 3 κ α2 + 3 κ γ2 + a2 k1 e ± 3 κ α2+ 3 κ γ2+ a2 + k2 ⎟⎟ 4(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎝ ⎠
202
(53)
Optik - International Journal for Light and Electron Optics 184 (2019) 197–204
Y. Yıldırım
3
3
1 2 2
2
4(κ α2+ κ γ2+ κ a2 + ω) ⎛ ± (x − ρt ) ⎞ 3 κ α2 + 3 κ γ2 + a2 3 κ α2+ 3 κ γ2+ a2 ⎜ ± 4(κ3α2 + κ3γ2 + κ 2a2 + ω) k1 e ⎟ ×⎜ ⎟ 4(κ3α2+ κ3γ2+ κ 2a2 + ω) (x − ρt ) ⎜⎜ 3 κ α2 + 3 κ γ2 + a2 k1 e ± 3 κ α2+ 3 κ γ2+ a2 + k2 ⎟⎟ 3 3 2 ⎝ 4(κ α2 + κ γ2 + κ a2 + ω) ⎠
⎫ ⎪ ⎪ i (−κx + ωt + ζ ) e ⎬ ⎪ ⎪ ⎭
(54)
and also if we set
k1 =
4(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
e
±
4(κ 3αj + κ 3γj + κ 2aj + ω) ϑ0 3 κ αj + 3 κ γj + aj
,
k2 = ± 1,
(55)
in Eqs. (53) and (54), the following strategic bright optical soliton in additionally singular optical soliton are given as 1 2
ψ (x , t ) =
3 3 2 ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎤⎫ ⎡ κ α1 + κ γ1 + κ a1 + ω sech2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β1 + κ η1 + b1 + c1 3 κ α1 + 3 κ γ1 + a1 ⎦⎭ ⎣ ⎩
(56)
1 2
ϕ (x , t ) =
3 3 2 ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎤⎫ ⎡ κ α2 + κ γ2 + κ a2 + ω sech2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β2 + κ η2 + b2 + c2 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(57)
The results (56) and (57) mean bright soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) > 0. 1 2
3 3 2 ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎤⎫ ⎡ κ α1 + κ γ1 + κ a1 + ω ψ (x , t ) = − csch2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β1 + κ η1 + b1 + c1 3 κ α1 + 3 κ γ1 + a1 ⎦⎭ ⎣ ⎩
(58)
1 2
3 3 2 ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎤⎫ ⎡ κ α2 + κ γ2 + κ a2 + ω ϕ (x , t ) = − csch2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β2 + κ η2 + b2 + c2 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(59)
The results (58) and (59) stems from singular soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) > 0. 1 2
κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎤⎫ ⎡ ψ (x , t ) = sec2 ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β1 + κ η1 + b1 + c1 3 κ α1 + 3 κ γ1 + a1 ⎦⎭ ⎣ ⎩
(60)
1 2
ϕ (x , t ) =
κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎤⎫ ⎡ sec2 ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β2 + κ η2 + b2 + c2 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(61)
1 2
ψ (x , t ) =
κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) 2 ⎡ ⎤⎫ csc ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎨ κ β1 + κ η1 + b1 + c1 ⎬ 3 κ α1 + 3 κ γ1 + a1 ⎣ ⎦⎭ ⎩
(62)
1 2
ϕ (x , t ) =
κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) 2 ⎡ ⎤⎫ csc ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎨ κ β2 + κ η2 + b2 + c2 ⎬ 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(63)
The results (60)–(63) mean singular periodic solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) < 0. 3. Conclusions Optical solitons with the Sasa-Satsuma equation in birefringent fibers have been worked by modified simple equation methodology in this manuscript. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton pulse, bright optical soliton pulse in additionally singular optical soliton pulse. An important point in obtaining these solutions is parameter restrictions because these parameters compensate these optical soliton pulses or molecules. The results from this article show us that the model can be improved further, for example, the Sasa-Satsuma equation can be extended to not only DWDM system but also birefringent fibers with FWM that can be viewed in many ways, for example, extended Kudryashov's technique, lie symmetry analysis, trial equation methodology, F -expansion scheme, modified simple equation approach 203
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Y. Yıldırım
to achieve optical soliton pulses. The results of these highly valuable studies will be reported in the optical or physics journals as soon as possible. References [1] A.R. Seadawy, A.H. Arnous, A. Biswas, M. Belic, Optical solitons with Sasa-Satsuma equation by F-expansion scheme, Optoelectron. Adv. Mater.-Rapid Commun. 13 (1–2) (2019) 31–36. [2] C. Gilson, J. Hietarinta, J. Nimmo, Y. Ohta, Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions, Phys. Rev. E 68 (1) (2003) 016614. [3] Y. Yıldırım, Optical solitons to Sasa-Satsuma model with trial equation approach, Optik (2019). [4] Y. Yıldırım, Optical solitons to Sasa-Satsuma model in birefringent fibers with trial equation approach, Optik (2019). [5] Y. Yıldırım, Optical solitons to Sasa-Satsuma model with modified simple equation approach, Optik (2019). [6] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. 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Belic, Optical soliton perturbation with Gerdjikov-Ivanov equation by modified simple equation method, Optik 157 (2018) 1235–1240. [11] A. Biswas, Y. Yildirim, E. Yasar, M.F. Mahmood, A.S. Alshomrani, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for Radhakrishnan-KunduLakshmanan equation with a couple of integration schemes, Optik 163 (2018) 126–136. [12] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Belic, Sub pico-second pulses in mono-mode optical fibers with Kaup-Newell equation by a couple of integration schemes, Optik 167 (2018) 121–128. [13] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, M.F. Mahmood, S.P. Moshokoa, M. Belic, Optical solitons with differential group delay for coupled Fokas-Lenells equation using two integration schemes, Optik 165 (2018) 74–86. [14] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton solutions to Fokas-lenells equation using some different methods, Optik 173 (2018) 21–31. [15] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Dispersive optical solitons with differential group delay by a couple of integration schemes, Optik 162 (2018) 108–120. [16] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with quadratic-cubic nonlinearity using a couple of strategic algorithms, Chin. J. Phys. 56 (5) (2018) 1990–1998. [17] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical solitons with differential group delay and four-wave mixing using two integration procedures, Optik 167 (2018) 170–188. [18] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Alfiras, M. Belic, Optical solitons in birefringent fibers with weak non-local nonlinearity using two forms of integration architecture, Optik 178 (2019) 669–680. [19] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S. Khan, S. Adesanya, S.P. Moshokoa, M. Belic, Optical soliton molecules in birefringent fibers having weak non-local nonlinearity and four-wave mixing with a couple of strategic integration architectures, Optik 179 (2019) 927–940. [20] 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. [21] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Chiral solitons with bohm potential by modified simple equation method and trial equation scheme, Acta Phys. Pol. A 134 (6) (2018) 1120–1125. [22] Y. Yıldırım, Sub pico-second pulses in mono-mode optical fibers with Triki-Biswas model using trial equation architecture, Optik 183 (2019) 463–466. [23] Y. Yildirim, Optical solitons of Biswas-Arshed equation by trial equation technique, Optik 182 (2019) 876–883. [24] Y. Yildirim, Optical solitons of Biswas-Arshed equation in birefringent fibers by trial equation technique, Optik 182 (2019) 810–820. [25] Y. Yildirim, Optical solitons of Biswas-Arshed equation by modified simple equation technique, Optik 182 (2019) 986–994. [26] Y. Yildirim, Optical solitons to Biswas-Arshed model in birefringent fibers using modified simple equation architecture, Optik 182 (2019) 1149–1162. [27] Y. Yıldırım, Optical solitons to Chen-Lee-Liu model in birefringent fibers with modified simple equation approach, Optik 183 (2019) 612–618. [28] Y. Yildirim, Optical solitons in DWDM system with trial equation integration architecture, Optik 182 (2019) 211–218. [29] Y. Yildirim, Optical solitons in DWDM technology with four-wave mixing by trial equation integration architecture, Optik 182 (2019) 625–632. [30] Y. Yildirim, Optical solitons to Schrödinger-Hirota equation in DWDM system with trial equation integration architecture, Optik 182 (2019) 275–281. [31] Y. 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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons to Sasa-Satsuma model in birefringent fibers with modified simple equation approach
T
Yakup Yıldırım 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: Modified simple equation architecture Sasa-Satsuma model Optical soliton pulses
Optical soliton pulses or molecules with the Sasa-Satsuma equation in birefringent fibers have been worked by modified simple equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.
1. Introduction Optical soliton pulses or molecules constitute of the basic fabric for soliton transmission technology such as transcontinental and transoceanic distances, data transmissionacross, optical fibers and telecommunications industry. The dynamics of these molecules or pulses have scuplted the technology to ionospherical level. These developments have prompted more comprehensive researches in this field from an physics and engineering aspect. One of the areas of influential research in this field is the work of the governing model along with fiber nonlinearities, for example, Lakshmanan–Porsezian–Daniel equation, Schrödinger equation with quadraticcubic nonlinearity, Kaup–Newell equation, Gerdjikov–Ivanov model, Schrödinger–Hirota equation, complex Ginzburg–Landau model, Radhakrishnan–Kundu–Lakshmanan equation, Fokas–Lenells equation, Schrödinger equation with weak non-local nonlinearity, Chen–Lee–Liu model, Triki-Biswas model, Biswas-Arshed model, DWDM system and many more [6–31] where soliton pulses along the one, two and N component of the models have been taken into consideration. Moreover, Sasa-Satsuma equation also is given as a model describing this phenomena. This model is an extension of Schrödinger equation which includes the self-steepening, third-order dispersion in additionally stimulated Raman scattering effects in monomode optical fibers. Additionally, this model describes the propagation of femtosecond pulses in optical fibers and also governs the propagation and interaction of the ultrashort pulses in the sub-picosecond or femtosecond regime. Because of this reason, finding optical soliton pulses or molecules to the model in birefringent fibers getting out of four-wave mixing terms (FWM) becomes very important. To achieve this goal, this article investigates the optical soliton pulses or molecules using modified simple equation architecture. In addition to such soliton type solutions, this method give rise to singular-periodic solutions. An important point in obtaining these solutions is parameter restrictions because these parameters compensate these optical soliton pulses or molecules. The detailed steps of obtaining the solutions of the aforementioned governing model are given the following chapters. 1.1. Governing model The Sasa-Satsuma governing model [1–5] can be given as follows
iψt + aψxx + b |ψ|2 ψ + i [αψxxx + β |ψ|2 ψx + θ (|ψ|2 )x ψ] = 0.
(1)
The temporal evolution of optical soliton molecules is given with the first term and also the Kerr law fiber nonlinearity is yield with the coefficient of b. Moreover, the group velocity dispersion term is provided with the coefficient of a and the profile of the optical https://doi.org/10.1016/j.ijleo.2019.03.022 Received 7 March 2019; Accepted 7 March 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 184 (2019) 197–204
Y. Yıldırım
soliton pulses is compensated by ψ(x, t). Lastly, the self-steepening, stimulated Raman scattering in additionally third-order dispersion sequentially are given with the coefficient of β, θ, α. The Sasa-Satsuma model in birefringent fibers without FWM to be discussed in this article is given as follows
iψt + a1 ψxx + (b1 |ψ|2 + c1 |ϕ|2 ) ψ + i [α1 ψxxx + γ1 ϕxxx + (β1 |ψ|2 + η1 |ϕ|2 ) ψx + (θ1 (|ψ|2 )x + λ1 (|ϕ|2 )x ) ψ] = 0, iϕt + a2 ϕxx + (b2 |ϕ|2 + c2 |ψ|2 ) ϕ
+ i [α2 ϕxxx + γ2 ψxxx + (β2 |ϕ|2 + η2 |ψ|2 ) ϕx + (θ2 (|ϕ|2 )x + λ2 (|ψ|2 )x ) ϕ] = 0
(2)
with the inclusion of the self-steepening, stimulated Raman scattering in additionally third-order dispersion that sequentially are given with the coefficient of βj, ηj, θj, λj, αj, γj. 2. A quick glance over modified simple equation technique In this chapter of the manuscript, iteration of the modified simple equation scheme [6–21] is given to work optical soliton molecules. Step-1: A nonlinear evolution equation used in the modeling of any physical event such as soliton transmission technology can be yield as (3)
V (r , rt , rx , rtt, rxt, rxx , …) = 0 and also this strategic equation can be collapsed into the following ordinary differential equation
F (P , P′, P ″, P″ ′, …) = 0
(4)
by using of
r (x , t ) = P (φ)
(5)
as long as
φ = x − ρt.
(6)
Step-2: The auxiliary equation corresponding to the ordinary differential equation can be given N
P (φ) =
Q′ (φ)
i
∑ δi ⎛ Q (φ) ⎞ i=0
⎜
⎟
⎝
⎠
(7)
having the essential constant coefficients δ0, δ1, …, δN where the N number comes from the balancing rule in Eq. (4). Step-3: The overdeterminet equations needed to find optical soliton pulses to Eq. (3) can be yield if Eq. (7) is put in Eq. (4) and subsequently the coefficients of Q−l are supposed as zero sequentially. Thus, the coefficients in the auxiliary equation are given if the overdeterminet equations can be solved. As a result, the optical soliton pulses of Eq. (3) can be yield by using of the obtained coefficients in the auxiliary equation. 2.1. Optical soliton pulses to the model In order to work optical soliton pulses with the Sasa-Satsuma equation, the following transformation can be yield as
ψ (x , t ) = P1 (ϑ) eiφ (x , t ) ϕ (x , t ) = P2 (ϑ) eiφ (x , t )
(8)
ϑ = x − ρt ,
(9)
with
φ (x , t ) = −κx + ωt + ζ .
(10)
The phase and amplitude component sequentially is given with the functions φ(x, t), Pj(ϑ) while the frequency, phase in additionally wave number respectively are given as the parameters κ, ζ, ω. The real and imaginary components sequentially can be yield as
(aj + 3κ αj ) P″j + 3κ γj P ″j˜ − (ω + κ 2aj + κ 3αj ) Pj − κ 3γj P j˜ + (bj + κ βj ) P 3j + (cj + κ ηj ) Pj P j2˜ = 0,
(11)
αj P″j ′ + γj P ″j˜ ′ − (2 κ aj + ρ + 3 κ 2αj ) P′j − 3κ 2γj P ′j˜ + (2 θj + βj ) P′j P j2 + ηj P′j P j2˜ + 2λj Pj P j˜ P ′j˜ = 0
(12)
for j = 1, 2 with j˜ = 3 − j if Eq. (8) is put in Eq. (2). The following condition can be yield as 198
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Y. Yıldırım
P j˜ = Pj
(13)
which give rise to
(aj + 3κ αj + 3κ γj ) P″j − (ω + κ 2aj + κ 3αj + κ 3γj ) Pj + (bj + κ βj + cj + κ ηj ) P 3j = 0, 2 θj + βj + ηj + 2λj
(αj + γj ) P″j − (2 κ aj + ρ + 3 κ 2αj + 3κ 2γj ) Pj +
3
(14)
P 3j = 0
(15)
because of the balancing rule for the imaginary and real parts (11) and (12). We can given the following strategic conditions
aj + 3κ αj + 3κ γj αj + γj
=
ω + κ 2aj + κ 3αj + κ 3γj 2 κ aj + ρ +
3 κ 2α
j
+
3κ 2γj
=
3(bj + κ βj + cj + κ ηj ) 2 θj + βj + ηj + 2λj
(16)
which give rise to
ω=
8 κ 3αj 2 + 16 κ 3αj γj + 8 κ 3γj 2 + 8 κ 2aj αj + 8 κ 2aj γj + 3 κ ρ αj + 3 κ ρ γj + 2 κ aj 2 + ρ aj
βj = −
αj + γj
,
6 κ αj λj + 6 κ αj θj + 6 κ γj λj + 6 κ γj θj + aj ηj + 2 aj λj + 2 aj θj − 3 αj bj − 3 αj cj − 3 bj γj − 3 cj γj aj
(17)
because the amplitude component Pj holds not only Eq. (14) but also Eq. (15). In order to obtain optical soliton molecules to Sasa-Satsuma governing model discussed in this study with modified simple equation scheme, Eq. (14) is considered in the rest of the article along with Eq. (17). Case-1: The auxiliary equation corresponding to the real component can be given
Q′ (ϑ) ⎞ Pj (ϑ) = δ0 + δ1 ⎛ ⎝ Q (ϑ) ⎠ ⎜
⎟
(18)
P 3j .
which comes from the balancing rule between the terms P″j and The overdetermined equations needed to find optical soliton pulses to Sasa-Satsuma governing model discussed in this paper can be yield as Q−3 coeff.:
δ1 (Q′)3 (κ βj δ12 + κ δ12ηj + bj δ12 + cj δ12 + 6 κ αj + 6 κ γj + 2 aj ) = 0, Q
−2
(19)
coeff.:
3δ1 (Q′)((−3 κ αj − 3 κ γj − aj ) Q″ + (κ βj δ0 δ1 + κ δ0 δ1 ηj + bj δ0 δ1 + cj δ0 δ1 ) Q′) = 0, Q
−1
(20)
coeff.:
δ1 ((3 κ αj + 3 κ γj + aj ) Q″ ′ + (−κ 3αj − κ 3γj + 3 κ βj δ0 2 + 3 κ δ0 2ηj
− κ 2aj + 3 bj δ0 2 + 3 cj δ0 2 − ω) Q′) = 0,
(21)
0
Q coeff.:
− κ 3αj δ0 − κ 3δ0 γj + κ βj δ03 + κ δ03ηj − κ 2aj δ0 + bj δ03 + cj δ03 − ω δ0 = 0, if Eq. (18) is put in Eq. (14) and then the coefficients of Q the auxiliary equation are given as
δ0 = ±
κ 3αj + κ 3γj + κ 2aj + ω κ βj + κ ηj + bj + cj
,
δ1 = ± −
−3
,Q
−2
,Q
−1
(22)
0
, Q are supposed as zero sequentially. Thus, the coefficients in
6 κ αj + 6 κ γj + 2 aj κ βj + κ ηj + bj + cj
(23)
and also we get
Q″ = ± −
Q″ ′ = −
2(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
2(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
Q′, (24)
Q′. (25)
If we use Eqs. (24) and (25), we get: 199
Optik - International Journal for Light and Electron Optics 184 (2019) 197–204
Y. Yıldırım
Q′ = ± −
3 κ αj + 3 κ γj + aj 2(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
± −
2(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
, (26)
and
Q=−
3 κ αj + 3 κ γj + aj 2(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
± −
2(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
+ k2
(27)
with k1 and k2 integration constants. The optical soliton pulses of the Sasa-Satsuma governing model can be yield as follows
κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎧ ψ (x , t ) = ± ± ⎨ κ β1 + κ η1 + b1 + c1 ⎩ 3
3
−
6 κ α1 + 6 κ γ1 + 2 a1 κ β1 + κ η1 + b1 + c1
2
2(κ α1+ κ γ + κ a1+ ω) ⎛ 3 κ α1 + 3 κ γ1 + a1 ± − 3 κ α + 31κ γ + a (x − ρt ) ⎞ ⎫ 1 1 1 ⎟ ⎪ i (−κx + ωt + ζ ) ⎜ ± − 2(κ3α1 + κ3γ1 + κ 2a1 + ω) k1 e ×⎜ , ⎟⎬e 3 3 2 2(κ α1+ κ γ + κ a1+ ω) 3 κ α1 + 3 κ γ1 + a1 ± − 3 κ α + 31κ γ + a (x − ρt ) ⎜− 3 1 1 1 + k2 ⎟ ⎪ k1 e 3 2 ⎠⎭ ⎝ 2(κ α1 + κ γ1 + κ a1 + ω)
κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎧ ϕ (x , t ) = ± ± ⎨ κ β2 + κ η2 + b2 + c2 ⎩ 3
3
−
(28)
6 κ α2 + 6 κ γ2 + 2 a2 κ β2 + κ η2 + b2 + c2
2
2(κ α + κ γ + κ a + ω) ⎛ 3 κ α2 + 3 κ γ2 + a2 ± − 3 κ2α + 32κ γ + a2 (x − ρt ) ⎞ ⎫ 2 2 2 ⎟ ⎪ i (−κx + ωt + ζ ) ⎜ ± − 2(κ3α2 + κ3γ2 + κ 2a2 + ω) k1 e ×⎜ ⎟⎬e 3 3 2 2(κ α + κ γ + κ a + ω) 3 κ α2 + 3 κ γ2 + a2 ± − 3 κ2α + 32κ γ + a2 (x − ρt ) ⎜− 3 2 2 2 + k2 ⎟ ⎪ k1 e 3 2 ⎠⎭ ⎝ 2(κ α2 + κ γ2 + κ a2 + ω)
(29)
and also if we set
k1 = −
2(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
e
± −
2(κ 3αj + κ 3γj + κ 2aj + ω) ϑ0 3 κ αj + 3 κ γj + aj
,
k2 = ± 1,
(30)
in Eqs. (28) and (29), the following strategic dark optical soliton in additionally singular optical soliton are given as
ψ (x , t ) = ±
ϕ (x , t ) = ±
κ 3α1 + κ 3γ1 + κ 2a1 + ω κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎤ ⎡ tanh ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎦ ⎣
(31)
κ 3α2 + κ 3γ2 + κ 2a2 + ω κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎤ ⎡ tanh ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎦ ⎣
(32)
The results (31) and (32) mean dark soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) < 0. ψ (x , t ) = ±
ϕ (x , t ) = ±
κ 3α1 + κ 3γ1 + κ 2a1 + ω κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎤ ⎡ coth ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎦ ⎣
(33)
κ 3α2 + κ 3γ2 + κ 2a2 + ω κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎤ ⎡ coth ⎢ − (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎦ ⎣
(34)
The results (33) and (34) mean singular soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) < 0. ψ (x , t ) = ± −
ϕ (x , t ) = ± −
3 3 2 κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎡ κ α1 + κ γ1 + κ a1 + ω ⎤ tan ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎣ ⎦
(35)
3 3 2 κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎡ κ α2 + κ γ2 + κ a2 + ω ⎤ tan ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎣ ⎦
(36)
200
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Y. Yıldırım
ψ (x , t ) = ± −
ϕ (x , t ) = ± −
3 3 2 κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎡ κ α1 + κ γ1 + κ a1 + ω ⎤ cot ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β1 + κ η1 + b1 + c1 2(3 κ α1 + 3 κ γ1 + a1 ) ⎣ ⎦
(37)
3 3 2 κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎤ ⎡ κ α2 + κ γ2 + κ a2 + ω cot ⎢ (x − ρt + ϑ0 ) ⎥ ei (−κx + ωt + ζ ) κ β2 + κ η2 + b2 + c2 2(3 κ α2 + 3 κ γ2 + a2 ) ⎦ ⎣
(38)
The consequences (35)–(38) mean singular periodic solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) > 0. Case-2: We can get the new real component as follows
(aj + 3κ αj + 3κ γj )(−(V ′j )2 + 2Vj V ′′j ) − 4(ω + κ 2aj + κ 3αj + κ 3γj ) V j2
+ 4(bj + κ βj + cj + κ ηj ) V 3j = 0 by using of Pj =
1 V j2
(39)
and moreover the auxiliary equation corresponding to this new equation can be given 2
Q′ (ϑ) ⎞ Q′ (ϑ) ⎞ Vj (ϑ) = δ0 + δ1 ⎛ + δ2 ⎛ Q (ϑ) ⎝ ⎠ ⎝ Q (ϑ) ⎠ ⎜
⎟
⎜
⎟
(40)
(V ′j )2 , V 3j
V 3j ,
or which comes from the balancing rule between the terms either Vj V ″j . The overdeterminet equations needed to find optical soliton pulses to Sasa-Satsuma governing model discussed in this paper can be yield as Q−6 coeff.:
4δ22 (Q′)6 (κ βj δ2 + κ δ2 ηj + 6 κ αj + 6 κ γj + bj δ2 + cj δ2 + 2 aj ) = 0, Q
−5
(41)
coeff.:
12δ2 (Q′) 4 ((−3 κ αj δ2 − 3 κ δ2 γj − aj δ2 ) Q″ + (κ βj δ1 δ2 + κ δ1 δ2 ηj + 3 κ αj δ1 + 3 κ δ1 γj + bj δ1 δ2 + cj δ1 δ2 + aj δ1) Q′) = 0, Q
−4
(42)
coeff.:
− (Q′)3 ((−12 δ22κ αj − 12 δ22κ γj − 4 δ22aj ) Q″ ′ + (54 κ αj δ1 δ2 + 54 κ δ1 δ2 γj + 18 aj δ1 δ2 ) Q″ + (4 κ 3αj δ22 + 4 κ 3δ22γj + 4 κ 2aj δ22 − 12 κ βj δ0 δ22 − 12 κ βj δ12δ2 − 12 κ δ0 δ22ηj
− 12 κ δ12δ2 ηj − 36 κ αj δ0 δ2 − 9 κ αj δ12 − 36 κ δ0 δ2 γj − 9 κ δ12γj − 12 bj δ0 δ22 − 12 bj δ12δ2 − 12 cj δ0 δ22 − 12 cj δ12δ2 + 4 ω δ22 − 12 aj δ0 δ2 − 3 aj δ12) Q′) = 0, Q
−3
(43)
coeff.:
− 2(Q′)2 ((−9 κ αj δ1 δ2 − 9 κ δ1 δ2 γj − 3 aj δ1 δ2 ) Q″ ′ + (30 κ αj δ0 δ2 + 6 κ αj δ12 + 30 κ δ0 δ2 γj + 6 κ δ12γj + 10 aj δ0 δ2 + 2 aj δ12) Q″ + (4 κ 3αj δ1 δ2 + 4 κ 3δ1 δ2 γj + 4 κ 2aj δ1 δ2
− 12 κ βj δ0 δ1 δ2 − 2 κ βj δ13 − 12 κ δ0 δ1 δ2 ηj − 2 κ δ13ηj − 6 κ αj δ0 δ1 − 6 κ δ0 δ1 γj
− 12 bj δ0 δ1 δ2 − 2 bj δ13 − 12 cj δ0 δ1 δ2 − 2 cj δ13 + 4 ω δ1 δ2 − 2 aj δ0 δ1) Q′) = 0, Q
−2
(44)
coeff.:
(12 κ αj δ0 δ2 + 6 κ αj δ12 + 12 κ δ0 δ2 γj + 6 κ δ12γj + 4 aj δ0 δ2 + 2 aj δ12) Q′Q″ ′ + (12 κ αj δ0 δ2 − 3 κ αj δ12 + 12 κ δ0 δ2 γj − 3 κ δ12γj + 4 aj δ0 δ2 − aj δ12)(Q′ ′)2
+ (−18 κ αj δ0 δ1 − 18 κ δ0 δ1 γj − 6 aj δ0 δ1 ) Q′Q″ + (−8 κ 3αj δ0 δ2 − 4 κ 3αj δ12
− 8 κ 3δ0 δ2 γj − 4 κ 3δ12γj − 8 κ 2aj δ0 δ2 − 4 κ 2aj δ12 + 12 κ βj δ0 2δ2 + 12 κ βj δ0 δ12
+ 12 κ δ0 2δ2 ηj + 12 κ δ0 δ12ηj + 12 bj δ0 2δ2 + 12 bj δ0 δ12 + 12 cj δ0 2δ2
+ 12 cj δ0 δ12 − 8 ω δ0 δ2 − 4 ω δ12)(Q′)2 = 0,
(45) 201
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Y. Yıldırım
Q−1 coeff.:
− 2δ0 δ1 ((−3 κ αj − 3 κ γj − aj ) Q″ ′ + (4 κ 3αj + 4 κ 3γj + 4 κ 2aj − 6 κ βj δ0
− 6 κ δ0 ηj − 6 bj δ0 − 6 cj δ0 + 4 ω) Q′) = 0,
(46)
Q0 coeff.:
− 4 κ 3αj δ0 2 − 4 κ 3δ0 2γj − 4 κ 2aj δ0 2 + 4 κ βj δ03 + 4 κ δ03ηj + 4 bj δ03 + 4 cj δ03 − 4 ω δ0 2 = 0,
(47) −l
if Eq. (40) is put in Eq. (39) and then the coefficients of Q equation are given as
δ0 = 0,
δ2 = −
2(3 κ αj + 3 κ γj + aj )
are supposed as zero sequentially. Thus, the coefficients in the auxiliary
,
κ βj + κ ηj + bj + cj
16(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω)
δ1 = ±
(κ βj + κ ηj + bj + cj )2
(48)
and also we get
Q″ = ±
Q″ ′ =
4(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
4(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
Q′, (49)
Q′. (50)
If we use Eqs. (49) and (50), we get:
Q′ = ±
3 κ αj + 3 κ γj + aj 4(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
±
4(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
, (51)
and
Q=
3 κ αj + 3 κ γj + aj 4(κ 3αj + κ 3γj + κ 2aj + ω)
k1 e
±
4(κ 3αj + κ 3γj + κ 2aj + ω) ϑ 3 κ αj + 3 κ γj + aj
+ k2
(52)
with k1 and k2 integration constants. The optical soliton pulses of the Sasa-Satsuma governing model can be yield as follows
16(3 κ α1 + 3 κ γ1 + a1 )(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎧ ψ (x , t ) = ± ⎨ (κ β1 + κ η1 + b1 + c1 )2 ⎩ 3
3
2
4(κ α1+ κ γ1+ κ a1+ ω) ⎛ ± (x − ρt ) ⎞ 3 κ α1 + 3 κ γ1 + a1 3 κ α1+ 3 κ γ1+ a1 ⎜ ± 4(κ3α1 + κ3γ1 + κ 2a1 + ω) k1 e ⎟ 2(3 κ α1 + 3 κ γ1 + a1 ) ×⎜ ⎟− κβ +κη +b +c 3 3 2 4(κ α1+ κ γ1+ κ a1+ ω) 1 1 1 1 (x − ρt ) ⎜⎜ 3 κ α1 + 3 κ γ1 + a1 k1 e ± 3 κ α1+ 3 κ γ1+ a1 + k2 ⎟⎟ 4(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎝ ⎠
3
3
1 2 2
2
4(κ α1+ κ γ1+ κ a1+ ω) ⎛ ± (x − ρt ) ⎞ 3 κ α1 + 3 κ γ1 + a1 3 κ α1+ 3 κ γ1+ a1 ⎜ ± 4(κ3α1 + κ3γ1 + κ 2a1 + ω) k1 e ⎟ ×⎜ ⎟ 4(κ3α1+ κ3γ1+ κ 2a1+ ω) (x − ρt ) ⎟ ⎜⎜ 3 κ α1 + 3 κ γ1 + a1 k1 e ± 3 κ α1+ 3 κ γ1+ a1 + k 2⎟ 3 3 2 ⎝ 4(κ α1 + κ γ1 + κ a1 + ω) ⎠
⎫ ⎪ ⎪ i (−κx + ωt + ζ ) , e ⎬ ⎪ ⎪ ⎭
16(3 κ α2 + 3 κ γ2 + a2 )(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎧ ϕ (x , t ) = ± ⎨ (κ β2 + κ η2 + b2 + c2 )2 ⎩ 3
3
2
4(κ α2+ κ γ2+ κ a2 + ω) ⎛ ± (x − ρt ) ⎞ 3 κ α2 + 3 κ γ2 + a2 3 κ α2+ 3 κ γ2+ a2 ⎜ ± 4(κ3α2 + κ3γ2 + κ 2a2 + ω) k1 e ⎟ 2(3 κ α2 + 3 κ γ2 + a2 ) ×⎜ ⎟− κβ +κη +b +c 4(κ3α2+ κ3γ2+ κ 2a2 + ω) 2 2 2 2 (x − ρt ) ⎜⎜ 3 κ α2 + 3 κ γ2 + a2 k1 e ± 3 κ α2+ 3 κ γ2+ a2 + k2 ⎟⎟ 4(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎝ ⎠
202
(53)
Optik - International Journal for Light and Electron Optics 184 (2019) 197–204
Y. Yıldırım
3
3
1 2 2
2
4(κ α2+ κ γ2+ κ a2 + ω) ⎛ ± (x − ρt ) ⎞ 3 κ α2 + 3 κ γ2 + a2 3 κ α2+ 3 κ γ2+ a2 ⎜ ± 4(κ3α2 + κ3γ2 + κ 2a2 + ω) k1 e ⎟ ×⎜ ⎟ 4(κ3α2+ κ3γ2+ κ 2a2 + ω) (x − ρt ) ⎜⎜ 3 κ α2 + 3 κ γ2 + a2 k1 e ± 3 κ α2+ 3 κ γ2+ a2 + k2 ⎟⎟ 3 3 2 ⎝ 4(κ α2 + κ γ2 + κ a2 + ω) ⎠
⎫ ⎪ ⎪ i (−κx + ωt + ζ ) e ⎬ ⎪ ⎪ ⎭
(54)
and also if we set
k1 =
4(κ 3αj + κ 3γj + κ 2aj + ω) 3 κ αj + 3 κ γj + aj
e
±
4(κ 3αj + κ 3γj + κ 2aj + ω) ϑ0 3 κ αj + 3 κ γj + aj
,
k2 = ± 1,
(55)
in Eqs. (53) and (54), the following strategic bright optical soliton in additionally singular optical soliton are given as 1 2
ψ (x , t ) =
3 3 2 ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎤⎫ ⎡ κ α1 + κ γ1 + κ a1 + ω sech2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β1 + κ η1 + b1 + c1 3 κ α1 + 3 κ γ1 + a1 ⎦⎭ ⎣ ⎩
(56)
1 2
ϕ (x , t ) =
3 3 2 ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎤⎫ ⎡ κ α2 + κ γ2 + κ a2 + ω sech2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β2 + κ η2 + b2 + c2 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(57)
The results (56) and (57) mean bright soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) > 0. 1 2
3 3 2 ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎤⎫ ⎡ κ α1 + κ γ1 + κ a1 + ω ψ (x , t ) = − csch2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β1 + κ η1 + b1 + c1 3 κ α1 + 3 κ γ1 + a1 ⎦⎭ ⎣ ⎩
(58)
1 2
3 3 2 ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎤⎫ ⎡ κ α2 + κ γ2 + κ a2 + ω ϕ (x , t ) = − csch2 ⎢ (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β2 + κ η2 + b2 + c2 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(59)
The results (58) and (59) stems from singular soliton solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) > 0. 1 2
κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) ⎤⎫ ⎡ ψ (x , t ) = sec2 ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β1 + κ η1 + b1 + c1 3 κ α1 + 3 κ γ1 + a1 ⎦⎭ ⎣ ⎩
(60)
1 2
ϕ (x , t ) =
κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) ⎤⎫ ⎡ sec2 ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎬ ⎨ κ β2 + κ η2 + b2 + c2 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(61)
1 2
ψ (x , t ) =
κ 3α1 + κ 3γ1 + κ 2a1 + ω ⎧ 2(κ 3α1 + κ 3γ1 + κ 2a1 + ω) 2 ⎡ ⎤⎫ csc ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎨ κ β1 + κ η1 + b1 + c1 ⎬ 3 κ α1 + 3 κ γ1 + a1 ⎣ ⎦⎭ ⎩
(62)
1 2
ϕ (x , t ) =
κ 3α2 + κ 3γ2 + κ 2a2 + ω ⎧ 2(κ 3α2 + κ 3γ2 + κ 2a2 + ω) 2 ⎡ ⎤⎫ csc ⎢ − (x − ρt ) ⎥ ei (−κx + ωt + ζ ) ⎨ κ β2 + κ η2 + b2 + c2 ⎬ 3 κ α2 + 3 κ γ2 + a2 ⎦⎭ ⎣ ⎩
(63)
The results (60)–(63) mean singular periodic solutions along with
(3 κ αj + 3 κ γj + aj )(κ 3αj + κ 3γj + κ 2aj + ω) < 0. 3. Conclusions Optical solitons with the Sasa-Satsuma equation in birefringent fibers have been worked by modified simple equation methodology in this manuscript. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton pulse, bright optical soliton pulse in additionally singular optical soliton pulse. An important point in obtaining these solutions is parameter restrictions because these parameters compensate these optical soliton pulses or molecules. The results from this article show us that the model can be improved further, for example, the Sasa-Satsuma equation can be extended to not only DWDM system but also birefringent fibers with FWM that can be viewed in many ways, for example, extended Kudryashov's technique, lie symmetry analysis, trial equation methodology, F -expansion scheme, modified simple equation approach 203
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Y. Yıldırım
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