Optical solitons to Sasa-Satsuma model with modified simple equation approach

Optik - International Journal for Light and Electron Optics 184 (2019) 271–276

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Original research article

Optical solitons to Sasa-Satsuma model 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 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 and many more [5–24]. 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 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 [1–4] 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 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 . https://doi.org/10.1016/j.ijleo.2019.03.020 Received 6 March 2019; Accepted 6 March 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 184 (2019) 271–276

Y. Yıldırım

2. A quick glance over modified simple equation technique In this chapter of the manuscript, iteration of the modified simple equation scheme [5–24] 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

V (r , rt , rx , rtt, rxt, rxx , …) = 0

(2)

and also this strategic equation can be collapsed into the following ordinary differential equation

F (P , P′, P′ ′, P′ ′′, …) = 0

(3)

by using of

r (x , t ) = P (φ)

(4)

as long as

φ = x − ρt.

(5)

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

⎜

⎟

⎝

⎠

(6)

having the essential constant coefficients δ0 , δ1, …, δN where the N number comes from the balancing rule in Eq. (3). Step-3: The overdeterminet equations needed to find optical soliton pulses to Eq. (2) can be yield if Eq. (6) is put in Eq. (3) 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. (2) 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 ) = P (ϑ) eiφ (x , t )

(7)

ϑ = x − ρt ,

(8)

φ (x , t ) = −κx + ωt + ζ .

(9)

with

The phase and amplitude component sequentially is given with the functions φ (x , t ), P (ϑ) while the frequency, phase in additionally wave number respectively are given as the parameters κ, ζ , ω. The real and imaginary components can be yield as

(3ακ + a) P′ ′ − (ακ 3 + aκ 2 + ω) P + (βκ + b) P 3 = 0,

αP′ ′ − (3ακ 2 + 2aκ + ρ) P +

(10)

β + 2θ 3 P =0 3

(11)

sequentially if Eq. (7) is put in Eq. (1). We can given the following strategic conditions

3(βκ + b) 3ακ + a ακ 3 + aκ 2 + ω = = α 3ακ 2 + 2aκ + ρ β + 2θ

(12)

which give rise to

b=

6ακθ + aβ + 2aθ , 3α

ω=

8 α 2κ 3 + 8 aα κ 2 + 2 a2κ + 3 α κ ρ + aρ α

(13)

because the amplitude component P holds not only Eq. (10) but also Eq. (11). In order to obtain optical soliton molecules to Sasa-Satsuma governing model discussed in this study with modified simple equation scheme, Eq. (10) is considered in the rest of the article along with Eq. (13). Case-1: The auxiliary equation corresponding to the real component can be given 272

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Q′ (ϑ) ⎞ P (ϑ) = δ0 + δ1 ⎛ ⎝ Q (ϑ) ⎠ ⎜

⎟

(14) ′

which comes from the balancing rule between the terms P′ and P 3. The overdeterminet 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 (β κ δ12 + bδ12 + 6 α κ + 2 a) = 0, Q−2

(15)

coeff.:

3δ1 Q′ (−(3 α κ + a) Q′ ′ + (β κ δ0 δ1 + bδ0 δ1 ) Q′) = 0,

(16)

Q−1 coeff.: − δ1 (−(3 α κ + a) Q′ ′′ + (α κ 3 − 3 β κ δ0 2 + aκ 2 − 3 bδ0 2 + ω) Q′) = 0,

(17)

Q 0 coeff.:

− α κ 3δ0 + β κ δ03 − aκ 2δ0 + bδ03 − ω δ0 = 0,

(18)

if Eq. (14) is put in Eq. (10) and then the coefficients of Q−3, Q−2, Q−1, Q 0 are supposed as zero sequentially. Thus, the coefficients in the auxiliary equation are given as

α κ 3 + aκ 2 + ω , βκ + b

δ0 = ±

δ1 = ± −

6α κ + 2a βκ + b

(19)

and also we get

Q′ ′ = ± −

Q′ ′′ = −

2(α κ 3 + aκ 2 + ω) Q′, 3α κ + a

(20)

2(α κ 3 + aκ 2 + ω) Q′. 3α κ + a

(21)

If we use Eqs. (20) and (21), we get:

Q′ = ± −

3α κ + a k1 e ± 2(α κ 3 + aκ 2 + ω)

−

2(α κ 3+ aκ 2+ ω) ϑ 3 α κ+a ,

(22)

and

Q=−

3α κ + a k1 e ± 2(α κ 3 + aκ 2 + ω)

−

2(α κ 3+ aκ 2+ ω) ϑ 3 α κ+a

+ k2

(23)

with k1 and k2 integration constants. The optical soliton pulses of the Sasa-Satsuma governing model can be yield as follows

⎧ ⎪ ψ (x , t ) = ± ⎨ ⎪ ⎩ i ( − κx + ωt +ζ ) ×e

3

α κ 3 + aκ 2 + ω βκ+b

6α κ+2a − βκ+b

±

2

⎛ ± − 3 α κ + a k e± − 2(α κ3 α+κaκ+ a+ ω) (x − ρt ) ⎞ ⎫ 1 2(α κ3 + aκ 2 + ω) ⎜ ⎟⎪ 2(α κ3+ aκ 2+ ω) ⎜ ⎟⎬ ± − (x − ρt ) 3α κ + a 3 α κ+a ⎜ − 3 2 k1 e + k2 ⎟ ⎪ ⎝ 2(α κ + aκ + ω) ⎠⎭ (24)

and also if we set

k1 = −

2(α κ 3 + aκ 2 + ω) ± e 3α κ + a

−

2(α κ 3+ aκ 2+ ω) ϑ0 3 α κ+a ,

k2 = ± 1,

(25)

in Eq. (24), the following strategic dark optical soliton in additionally singular optical soliton are given as

ψ (x , t ) = ±

α κ 3 + aκ 2 + ω α κ 3 + aκ 2 + ω i (−κx + ωt + ζ ) tanh ⎡ (x − ρt + ϑ0 ) ⎤ − ⎢ ⎥e βκ + b 2(3 α κ a ) + ⎣ ⎦

(26)

The result (26) means dark soliton solutions along with

(3ακ + a)(ακ 3 + aκ 2 + ω) < 0.

ψ (x , t ) = ±

α κ 3 + aκ 2 + ω α κ 3 + aκ 2 + ω ⎤ i (−κx + ωt + ζ ) coth ⎡ ⎢ − 2(3 α κ + a) (x − ρt + ϑ0 ) ⎥ e βκ + b ⎣ ⎦ 273

(27)

Optik - International Journal for Light and Electron Optics 184 (2019) 271–276

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The result (27) means singular soliton solutions along with

(3ακ + a)(ακ 3 + aκ 2 + ω) < 0.

ψ (x , t ) = ± −

ψ (x , t ) = ± −

α κ 3 + aκ 2 + ω α κ 3 + aκ 2 + ω i (−κx + ωt + ζ ) tan ⎡ (x − ρt + ϑ0 ) ⎤ ⎢ ⎥e βκ + b 2(3 α κ + a) ⎣ ⎦

(28)

α κ 3 + aκ 2 + ω α κ 3 + aκ 2 + ω i (−κx + ωt + ζ ) cot ⎡ (x − ρt + ϑ0 ) ⎤ ⎢ ⎥e βκ + b 2(3 α κ a ) + ⎣ ⎦

(29)

The consequences (28) and (29) mean singular periodic solutions along with

(3ακ + a)(ακ 3 + aκ 2 + ω) > 0. Case-2: We can get the new real component as follows

(3ακ + a)(−(V ′)2 + 2VV′ ′) − 4(ακ 3 + aκ 2 + ω) V 2 + 4(βκ + b) V 3 = 0 1 V2

by using of P =

(30)

and moreover the auxiliary equation corresponding to this new equation can be given 2

Q′ (ϑ) ⎞ Q′ (ϑ) ⎞ V (ϑ) = δ0 + δ1 ⎛ + δ2 ⎛ ⎝ Q (ϑ) ⎠ ⎝ Q (ϑ) ⎠ ⎜

⎟

⎜

⎟

(31)

(V ′)2 ,

V3

V 3,

′

VV′ . or which comes from the balancing rule between the terms either 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 (β κ δ2 + 6 α κ + bδ2 + 2 a) = 0,

Q−5

(32)

coeff.:

12δ2 (Q′) 4 ((−3 α κ δ2 − aδ2 ) Q′ ′ + (β κ δ1 δ2 + 3 α κ δ1 + bδ1 δ2 + aδ1 ) Q′) = 0,

Q−4

(33)

coeff.:

− (Q′)3 ((−12 α κ δ22 − 4 aδ22) Q′ ′′ + (54 α κ δ1 δ2 + 18 aδ1 δ2 ) Q′ ′ + (4 α κ 3δ22 + 4 aκ 2δ22 − 12 β κ δ0 δ22 − 12 β κ δ12δ2 − 36 α κ δ0 δ2 − 9 α κ δ12 − 12 bδ0 δ22 − 12 bδ12δ2 − 12 aδ0 δ2 − 3 aδ12 + 4 ω δ22) Q′)=0,

(34)

Q−3 coeff.: − 2(Q′)2 ((−9 α κ δ1 δ2 − 3 aδ1 δ2 ) Q′ ′′ + (30 α κ δ0 δ2 + 6 α κ δ12 + 10 aδ0 δ2 + 2 aδ12) Q′ ′ + (4 α κ 3δ1 δ2 + 4 aκ 2δ1 δ2 − 12 β κ δ0 δ1 δ2 − 2 β κ δ13 − 6 α κ δ0 δ1 − 12 bδ0 δ1 δ2 − 2 bδ13 − 2 aδ0 δ1 + 4 ω δ1 δ2) Q′)=0, Q−2

(35)

coeff.:

(12 α κ δ0 δ2 + 6 α κ δ12 + 4 aδ0 δ2 + 2 aδ12) Q′Q′ ′′ + (12 α κ δ0 δ2 − 3 α κ δ12 + 4 aδ0 δ2 − aδ12)(Q′ ′)2 + (−18 α κ δ0 δ1 − 6 aδ0 δ1 ) Q′Q′ ′ + (−8 α κ 3δ0 δ2 − 4 α κ 3δ12 − 8 aκ 2δ0 δ2 − 4 aκ 2δ12 + 12 β κ δ0 2δ2 + 12 β κ δ0 δ12 + 12 bδ0 2δ2 + 12 bδ0 δ12 − 8 ω δ0 δ2 − 4 ω δ12)(Q′)2 = 0, Q−1

coeff.:

− δ0 δ1 ((−3 α κ − a) Q′ ′′ + (4 α κ 3 + 4 aκ 2 − 6 β κ δ0 − 6 bδ0 + 4 ω) Q′) = 0, Q0

(36)

(37)

coeff.:

− 4 α κ 3δ0 2 − 4 aκ 2δ0 2 + 4 β κ δ03 + 4 bδ03 − 4 ω δ0 2 = 0, if Eq. (31) is put in Eq. (30) and then the coefficients of equation are given as

δ0 = 0,

δ1 = ±

16(3 α κ + a)(ακ 3 + aκ 2 + ω) , (β κ + b)2

Q−l

(38)

are supposed as zero sequentially. Thus, the coefficients in the auxiliary

δ2 = −

2(3 α κ + a) βκ + b

and also we get 274

(39)

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Y. Yıldırım

4(α κ 3 + aκ 2 + ω) Q′, 3α κ + a

Q′ ′ = ± Q′ ′′ =

(40)

4(α κ 3 + aκ 2 + ω) Q′. 3α κ + a

(41)

If we use Eqs. (40) and (41), we get:

Q′ = ±

3α κ + a k1 e ± 4(α κ 3 + aκ 2 + ω)

4(α κ 3+ aκ 2+ ω) ϑ 3 α κ+a ,

(42)

and

Q=

3α κ + a k1 e ± 4(α κ 3 + aκ 2 + ω)

4(α κ 3+ aκ 2+ ω) ϑ 3 α κ+a

+ k2

(43)

with k1 and k2 integration constants. The optical soliton pulses of the Sasa-Satsuma governing model can be yield as follows

⎧ ⎪ ψ (x , t )= ± ⎨ ⎪ ⎩

3

16(3 α κ + a)(ακ 3 + aκ 2 + ω) (β κ + b)2

2

4(α κ + aκ + ω) ⎛± (x − ρt ) ⎞ ± 3α κ + a 3 α κ+a k1 e ⎟ ⎜ 4(α κ3 + aκ2 + ω) 4(α κ3+ aκ 2+ ω) ⎟ ⎜ 3α κ + a (x − ρt ) ± 3 α κ+a ⎜ k e + k2 ⎟ 3 + aκ 2 + ω) 1 4( α κ ⎠ ⎝ 3

2

1 2 2

4(α κ + aκ + ω) ⎛± (x − ρt ) ⎞ ± 3α κ + a 3 α κ+a k1 e 4(α κ3 + aκ 2 + ω) 2(3 α κ + a) ⎜ ⎟ − βκ+b 4(α κ3+ aκ 2+ ω) ⎟ ⎜ 3α κ + a (x − ρt ) ± 3 α κ + a ⎜ k1 e + k2 ⎟ 3 2 ⎠ ⎝ 4(α κ + aκ + ω)

⎫ ⎪ ⎬ ⎪ ⎭

ei (−κx + ωt + ζ ) (44)

and also if we set

k1 =

4(α κ 3 + aκ 2 + ω) ± e 3α κ + a

4(α κ 3+ aκ 2+ ω) ϑ0 3 α κ+a ,

k2 = ± 1,

(45)

in Eq. (44), the following strategic bright optical soliton in additionally singular optical soliton are given as 1 2 α κ 3 + aκ 2 + ω ⎧ 2(ακ 3 + aκ 2 + ω) ⎫ i (−κx + ωt + ζ ) ψ (x , t ) = sech2 ⎡ (x − ρt ) ⎤ ⎢ ⎥⎬ e ⎨ βκ + b 3 α κ a + ⎣ ⎦⎭ ⎩

(46)

The result (46) means bright soliton solutions along with

(3ακ + a)(ακ 3 + aκ 2 + ω) > 0. 1 2 α κ 3 + aκ 2 + ω ⎧ 2(ακ 3 + aκ 2 + ω) ⎫ i (−κx + ωt + ζ ) ψ (x , t ) = − csch2 ⎡ (x − ρt ) ⎤ e ⎢ ⎥ ⎨ ⎬ βκ + b 3α κ + a ⎣ ⎦⎭ ⎩

(47)

The result (47) stems from singular soliton solutions along with

(3ακ + a)(ακ 3 + aκ 2 + ω) > 0. 1

ψ (x , t ) =

2 α κ 3 + aκ 2 + ω ⎧ 2(ακ 3 + aκ 2 + ω) 2 ⎡ ⎫ i (−κx + ωt + ζ ) sec ⎢ − (x − ρt ) ⎤ e ⎥ ⎨ ⎬ βκ + b 3α κ + a ⎣ ⎦⎭ ⎩

(48)

1 2 α κ 3 + aκ 2 + ω ⎧ 2(ακ 3 + aκ 2 + ω) 2 ⎡ ⎫ i (−κx + ωt + ζ ) ψ (x , t ) = csc ⎢ − (x − ρt ) ⎤ ⎥⎬ e ⎨ βκ + b 3α κ + a ⎣ ⎦⎭ ⎩

(49)

The results (48)-(49) mean singular periodic solutions along with

(3ακ + a)(ακ 3 + aκ 2 + ω) < 0. 3. Conclusions Optical solitons with the Sasa-Satsuma equation 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 275

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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 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 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, (2019) Optik. [4] Y. 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