key: cord-0908736-y4zwd6wg
authors: Yalçı, Ali Murat; Ekici, Mehmet
title: Stationary optical solitons with complex Ginzburg–Landau equation having nonlinear chromatic dispersion
date: 2022-02-16
journal: Opt Quantum Electron
DOI: 10.1007/s11082-022-03557-3
sha: 5454d7ebd017dd3b0e86014078daec94f27e8821
doc_id: 908736
cord_uid: y4zwd6wg

The current work is on the retrieval of stationary soliton solutions to the complex Ginzburg–Landau equation that is studied with nonlinear chromatic dispersion having a plethora of nonlinear refractive index structures. The Jacobi’s elliptic function approach is employed to recover doubly periodic waves which leads to soliton solutions when the limiting value of the modulus of ellipticity is reached.

The analytics and the rapid advancement of the technology of optical solitons have portrayed a lasting impression in the telecommunications industry Triki et al. 2012; Mirzazadeh et al. 2016; Arnous et al. 2017; Biswas and Alqahtani 2017; Biswas et al. 2018a, b; Biswas 2018; Arshed et al. 2019; Das et al. 2019; Zayed et al. 2020 Zayed et al. , 2021 Yan et al. 2020; Biswas et al. , 2012 Mirzazadeh et al. 2014; Liu et al. 2018; Biswas et al. 2016; Biswas and Arshed 2018; Liu et al. 2019; Bakodah et al. 2017; Zhou et al. 2014; Adem et al. , 2021 Atai and Malomed 2001; Biswas and Konar 2006; Khalique 2011, 2013; Ekici et al. 2018 Ekici et al. , 2021 Geng and Li 2008; Guo and Zhou 2010; Kara 2021; Kudryashov 2019 Kudryashov , 2020a Kudryashov , b, c, d, e, f, 2021a Sonmezoglu et al. 2021; Sucu et al. 2021; Susanto and Malomed 2021; Yan 2006a, b; Zhang et al. 2010; Zhou et al. 2016; Zayed 2009; Malik et al. 2012 ). This gave way to a plethora of results and uncountable avenues for performance enhancement in this field. There are several models that govern the dynamical flow and of solitons across inter-continental distances. While the most visible model is the nonlinear Schrödinger's equation, it is often necessary to veer off to other models depending on the circumstantial situation. For example, dispersive solitons are governed by Schrödinger-Hirota equation or Fokas-Lenells equation and others.

Today's paper will address the complex Ginzburg-Landau equation (CGLE) Triki et al. 2012; Mirzazadeh et al. 2016; Arnous et al. 2017; Biswas and Alqahtani 2017; Biswas et al. 2018a, b; Biswas 2018; Arshed et al. 2019; Das et al. 2019; Zayed et al. 2020; Yan et al. 2020; Zayed et al. 2021; ) that is also an alternative model that governs the soliton propelling dynamics for long distances. This model is studied with nonlinear chromatic dispersion (CD) . Several forms of self-phase modulation (SPM) structures (Biswas and Konar 2006) are studied in the paper. Rough handling of fibers and other issues, such as environmental causes, may lead to CD being rendered nonlinear. In such a situation the solitons would become stationary and thus the information transfer for trans-continental and trans-oceanic distances would completely stall. This would lead to a catastrophic effect especially during COVID-19 times when the world is totally dependent on Internet activities. The analytical derivation of these stationary solitons for an abundant variety of SPM are displayed in the rest of the work. The stationary solitons are derived through an intermediary Jacobi's elliptic functions that approach soliton solutions when the modulus of ellipticity approaches its appropriate limit. The details are exhibited in the rest of the paper after a succinct intro to the model.

The dimensionless form of CGLE with nonlinear CD reads as Triki et al. 2012; Mirzazadeh et al. 2016; Arnous et al. 2017; Biswas and Alqahtani 2017; Biswas et al. 2018a Biswas et al. , 2018b Biswas 2018; Arshed et al. 2019; Das et al. 2019; Zayed et al. 2020; Yan et al. 2020; Zayed et al. 2021; where a, b, , and are constants and F stands for the nonlinear function. The first term stands for the linear evolution term, while the coefficient of a is the nonlinear CD and the third term accounts for the generalized nonlinear term. Next, the terms with , and arise from the perturbation effects; in particular comes from the detuning effect. Also, in the model (1), the independent variables are x and t which are spatial and temporal coordinates. The dependent variable q(x, t) is a complex-valued function which stands for the wave profile, q * (x, t) denotes the conjugate of q(x, t) and finally i = √ −1.

To extract stationary solutions to (1), initial assumption (Adem et al. , 2021 Khalique 2011, 2013; Sonmezoglu et al. 2021; Sucu et al. 2021) (1) iq t + a(|q| n q) xx + bF |q| 2 q = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2

x 2 + q Page 3 of 30 167 is considered. Here the constant is the wave number. Substituting (2) into (1), it is reached that Equation (3) will now be analyzed according to the type of nonlinear media in next subsections.

For this nonlinearity, Generalized (G � ∕G)-expansion approach will now be applied to deal with (7). To kick off, suppose Eq. (7) possess the solution as (Zayed 2009; Malik et al. 2012) where G = G(x) holds that is called Jacobi elliptic (JE) equation. Here i , e 0 , e 1 and e 2 are the arbitrary constants that need to be fixed such that n ≠ 0 . The solutions of Eq. (9) are presented as follows (Zayed 2009; Malik et al. 2012) :

Here, the modulus of JE functions is stood for by k (0 < k < 1) and i = √ −1. Balancing 4 with ′ 2 or 2 ′′ leads to N = 2 . Then Eq. (8) becomes

Inserting (10) along with (9) into (7), one recovers a polynomial in G j , G ′ G j (j = ±1, ±2, …) . Equating each coefficient of the polynomial obtained to zero and then overcoming the resulting systems yields Substituting (11) into (10) and employing (2) gives

Next, solutions for the model under consideration (6) are attained as below: If e 0 = 1 , e 1 = − k 2 + 1 , e 2 = k 2 ,

or For e 0 = 1 − k 2 , e 1 = 2k 2 − 1 , e 2 = −k 2 , When e 0 = k 2 − 1 , e 1 = 2 − k 2 , e 2 = −1,

In the case of e 0 = 1 , e 1 = 2k 2 − 1 , e 2 = k 2 k 2 − 1 , For the case e 0 = k 2 4 , e 1 = 1 2 k 2 − 2 , e 2 = 1 4 ,

Here, the solutions from (14) to (22) represent JE function solutions to the model. Next, for e 0 = 0 , e 1 = 1 , e 2 = −1 , dark soliton ( Fig. 1) is

q(x, t) = − 4a 2k 2 − 1 cd 4 x ns 2 x exp i − + 5 2k 2 − 1 + 12 k 2 k 2 − 1

When e 0 = 0 , e 1 = 1 , e 2 = 1 , singular soliton is Finally, if e 0 = 0 , e 1 = −1 , e 2 = 1 , periodic solution is Similarly, plugging (12) into (10) and utilizing (2) gives and then one gets the following solutions: For e 0 = 0 , e 1 = 1 , e 2 = −1 , bright soliton ( Fig. 2) is

If e 0 = 0 , e 1 = 1 , e 2 = 1 , other type of singular soliton emerges as When e 0 = 0 , e 1 = −1 , e 2 = 1 , periodic wave is Proceeding as in the case of Kerr law, one has the solution set as:

Substituting (37) into (36) and employing (2) gives and thus, the solutions to (32) are derived as:

Whenever e 0 = 0 , e 1 = 1 , e 2 = −1,

Finally, for e 0 = 0 , e 1 = 1 , e 2 = 1,

Here, the solutions (40)-(42) stands for JE function solutions, while the solutions (43) and (44) are respectively dark and singular solitons.

For this law, with the constants b 1 and b 2 . Then (1) 

Finally, when e 0 = 0 , e 1 = 1 , e 2 = 1,

Here, JE function solutions are represented by Eqs. (52)-(55), while dark and singular solitons are respectively indicated in Eqs. (56) and (57).

This law occurs when with the constants b 1 and b 2 . Thus, (1) changes to

For the integration of Eq. (59), n = 2m is selected. Thus, (59) simplifies to

and (3) (64) and utilizing (2) leads to and as a consequence, Eq. (60) possess the following solutions: 

This nonlinear form arises when with the constants b 1 and b 2 . Thus, the model (1) 

Balance principle causes N = 2 . Then Eq. (8) becomes Proceeding as in previous sections, one secures two solution set as

Substituting (78) into (77) 

In the case of e 0 = 0 , e 1 = 1 , e 2 = 1,

Here, the solutions given by Eqs. (81) Here, bright and singular solitons are respectively given by Eqs. (88) and (89), while periodic wave is given Eq. (90).

In the case of this law, Then Eq. (1) changes to and Eq. (3) modifies to

(91) F(s) = ln s.

(92) iq t + a(|q| n q) xx + 2bq ln |q| = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2

x 2 + q When e 0 = 0 , e 1 = −1 , e 2 = 1, (93) − ( + ) 2 + 2b 2 ln | | − 2( − 2 ) � 2 + an(n + 1) n � 2 − 2 �� + a(n + 1) n+1 �� = 0.

In Since b 1 = 0 from the solution set (109), this form of the nonlinearity collapses to parabolic law nonlinear media. Also, because the solution (110) is the same as that of in case of parabolic law. Hence, the solutions that will be recovered are omitted.

Page 17 of 30 167

For nonlinear form where b j for j = 1, 2, 3 are constants. Therefore, (1) When e 0 = k 2 − 1 , e 1 = 2 − k 2 , e 2 = −1,

Whenever e 0 = 0 , e 1 = 1 , e 2 = −1, Finally, when e 0 = 0 , e 1 = 1 , e 2 = 1,

Here, JE function solutions are represented by from (118) to (121), while dark and singular solitons are introduced in Eqs. (122) and (123), respectively. Putting (131) 

(121) q(x, t) = − 1 k 2 sd x cn x exp i 4 2 − k 2 − t . (122) q(x, t) = − 1 tanh x exp [i(4 − )t]. (123) q(x, t) = − 1 coth x exp [i(4 − )t]. (124) F(s) = b 1 s m + b 2 s 2m + b 3 s 3m (125) iq t + a(|q| n q) xx + b 1 |q| 2m + b 2 |q| 4m + b 3 |q| 6m q = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2 x 2 + q.(126)iq t + a |q| 4m q xx + b 1 |q| 2m + b 2 |q| 4m + b 3 |q| 6m q = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2 x 2 + q (127) − ( + ) 2 + b 1 2m+2 + b 2 4m+2 + b 3 6m+2 − 2( − 2 ) � 2 + 4am(4m + 1) 4m � 2 − 2 �� + a(4m + 1) 4m+1 �� = 0. (128) = 1 m (129) − m 2 ( + ) 2 + b 1 m 2 4 + b 2 m 2 6 + b 3 m 2 8 + 2( (m − 2) + 2 ) � 2 + a 12m 2 + 7m + 1 4 � 2 − 2m �� + am(4m + 1) 5 �� = 0. (130) (x) = 0 + 1 G � G

This nonlinear media arises when iq t + a(|q| n q) xx + b 1 |q| 2 + b 2 |q| 4 + b 3 |q| 2 xx q = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2

x 2 + q.

For e 0 = 1 − k 2 , e 1 = 2k 2 − 1 , e 2 = −k 2 ,

Here, the solutions (147) 

For this nonlinear media, 

Substituting (166) into (165) and employing (2) gives and thus, the solutions to (163) are listed as:

or For e 0 = 1 − k 2 , e 1 = 2k 2 − 1 , e 2 = −k 2 , When e 0 = k 2 − 1 , e 1 = 2 − k 2 , e 2 = −1,

Whenever e 0 = 0 , e 1 = 1 , e 2 = −1,

b 1 = − 8a 3 2 e 3 1 − 36e 0 e 1 e 2 2 (m + 2) 81 e 2 1 + 12e 0 e 2 (m + 1) 2 , b 3 = − 2a(m + 2)(3m + 5) 2 (m + 1) 2 , 0 = − 2 2 e 1 3 , 1 = 0, = a 2 e 1 e 2 1 − 36e 0 e 2 (m + 2) 9 e 2 1 + 12e 0 e 2 (m + 1) , = a 2 e 1 e 2 1 − 36e 0 e 2 (m + 2)(m + 5) 36 e 2 1 + 12e 0 e 2 (m + 1) , = − − 2a 2 e 2 1 + 12e 0 e 2 (m + 2)(m + 3) 3(m + 1) 2 . Similarly, putting (167) into (165) and using (2) leads to and thus, one acquires bright and singular solitons and also periodic wave solution, respectively as: If e 0 = 0 , e 1 = 1 , e 2 = −1,

For e 0 = 0 , e 1 = 1 , e 2 = 1,

When e 0 = 0 , e 1 = −1 , e 2 = 1,

For CQ nonlinearity, with the constants b 1 and b 2 . Thus Eq. (1) 

(178) q(x, t) = 2 sec 2 x 1 m+1 e i t .

iq t + a |q| 2 q xx + b 1 |q| 2 + b 2 |q| 3 q = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2

x 2 + q

−( + ) 2 + b 1 4 + b 2 5 − 2( − 2 ) � 2 + 6a 2 � 2 − 2 �� + 3a 3 �� = 0. and then one obtains bright and singular solitons and also periodic wave, respectively as: For e 0 = 0 , e 1 = 1 , e 2 = −1,

If e 0 = 0 , e 1 = 1 , e 2 = 1,

When e 0 = 0 , e 1 = −1 , e 2 = 1, iq t + a(|q| n q) xx + b 1 |q| 2m + b 2 |q| 3m q = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2

x 2 + q.

(199) iq t + a |q| 2m q xx + b 1 |q| 2m + b 2 |q| 3m q = 1 |q| 2 q * |q| 2 |q| 2 xx − |q| 2

x 2 + q

− ( + ) 2 + b 1 2m+2 + b 2 3m+2 − 2( − 2 ) � 2 + 2am(2m + 1) 2m � 2 − 2 �� + a(2m + 1) 2m+1 �� = 0.

(201) = 2 m (202) − m 2 ( + ) 2 + b 1 m 2 6 + b 2 m 2 8 + 4( (m − 4) + 4 ) � 2 + 2a 6m 2 + 7m + 2 4 � 2 − 4m �� + 2am(2m + 1) 5 �� = 0.

(203) (x) = 0 + 1 G � G Utilizing (204) into (203) and using (2) 

This work is on the derivation and exhibition of stationary solitons that emerged from CGLE that is with nonlinear CD and having several forms of SPM structures. Jacobi's elliptic functions approach has made this retrieval possible. The results are exhibited for linear temporal evolution. This paper has immediate follow-ups from several avenues. An instantaneous consequence of this paper would be to study the same model with generalized temporal evolution. This would give a generalized perspective to the model handled and studied here. Later the model would be handled numerically such as with the usage of variational iteration method, Adomian decomposition scheme and several others. Such results are yet to be reported and are currently awaited.

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The authors have not disclosed any funding.

The authors declare that there is no conflict of interest.

and then the results given below are derived: Substituting (184) into (183) and employing (2) gives As a results, JE function solutions, dark and singular solitons to the model (181) are written down as:or For e 0 = 1 − k 2 , e 1 = 2k 2 − 1 , e 2 = −k 2 , When e 0 = k 2 − 1 , e 1 = 2 − k 2 , e 2 = −1,Similarly, inserting (185) into (183) and employing (2) brings about= 45 2 2 a e 2 1 − 4e 0 e 2 16 , = 30a 2 2 e 1 e 2 1 − 4e 0 e 2 − .(192) q(x, t) = 2 coth 2 x exp i 30a 2 2 − t .(193) q(x, t) = − 2 e 1 + 2 G � G Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.