 Research
 Open Access
 Published:
Optical soliton perturbation of FokasLenells equation by the LaplaceAdomian decomposition algorithm
Journal of the European Optical SocietyRapid Publications volume 15, Article number: 13 (2019)
Abstract
This paper displays numerical simulation for bright and dark optical solitons that emerge from FokasLenells equation which is studied in the context of dispersive solitons in polarizationpreserving fibers. The LaplaceAdomian decomposition scheme is the numerical tool adopted in the paper. The numerical results, for bright and dark solitons, are expository and therefore supplement the analytical developments, thus far.
Introduction
One of the governing models to study dispersive solitons is FokasLennels equation (FLE) [1–13]. In such a model, in addition to group velocity dispersion (GVD), one considers, intermodal dispersion as well as nonlinear dispersion thus treating it with a flavor of additional dispersive effects. There has been a plethora of analytical tools that have been implemented to study FLE. They range from semiinverse variational principle, Lie symmetry analysis, Riccati equation approach, expfunction method, traveling wave hypothesis, trial function method and further wide varieties. This paper will be changing gears to study the model from a different perspective. One of the very many and modern numerical algorithms that will be implemented is the LaplaceAdomian decomposition integration scheme. This method has been successfully applied to variety of other models from optics [14–16]. This paper now studies FLE, for the first time, by the aid of LaplaceAdomian decomposition scheme. The details are sketched in the remainder of the paper, after introducing the model.
The FokasLenells equation (FLE) in presence of perturbation terms
The dimensionless form of the perturbed FokasLenells equation (FLE) is given by
This equation was first studied in [17–24] and arises in various systems such as water waves, plasma physics, solid state physics and nonlinear optics. In Eq. (1), u(x,t) represents a complex field envelope, and x and t are spatial and temporal variables, respectively. Here, the coefficient a_{1} is the group velocity dispersion (GVD) and a_{2} is the spatiotemporal dispersion (STD) the coefficient b is selfphase modulation moreover σ accounts for nonlinear dispersion. In the perturbative term of Eq. (1), the first term represents the intermodal dispersion (IMD), the second term is the selfsteepening effect and finally the last term accounts for another version of nonlinear dispersion (ND).
Bright optical solitons
The bright optical soliton solution to (1) is given by [5, 11]:
Here, ν is the soliton velocity, κ is the soliton frequency, ω is the angular velocity and θ is the phase center.
The amplitude A of the soliton in this case is given by
where, the velocity of the soliton in relation to the coefficients that appear in the Eq. (1) is
and the constraints conditions on the parameters are
In the previous context κ is any parameter that satisfies the Eq. (5).
Dark optical solitons
noindent The dark optical soliton solution to (1) is given by [5, 11]:
Here, ν is the soliton velocity, κ is the soliton frequency, ω is the angular velocity and θ is the phase center.
The amplitude B of the soliton in this case is given by
where, the velocity of the soliton in relation to the coefficients that appear in the Eq. (1) is
and the constraints conditions on the parameters are
In the previous context κ is any parameter that satisfies the Eq. (9).
The Laplace Adomian Decomposition Method (LADM)
To illustrate the basic concept of LaplaceAdomian decomposition algorithm, we consider the general form of second order nonlinear partial differential equations in the form
with initial conditions
where F is a differential operator. Now, let us decompose this operator as F=L+R+N where \(L(u)=\frac {\partial u}{\partial t}\) stands for a linear differential operator. The operators R and N are the remaining linear and nonlinear parts, respectively.
With these considerations, Eq. (10) can now be rewritten as
Solving for Lu(x,t) and applying the Laplace transform respect to t to Eq. (12), gives
Thus, Eq. (13) turns out to be equivalent to
Using Eq. (11), one get
Finally, by applying inverse Laplace transformation \(\mathcal {L}^{1}\) on both sides of the Eq. (15), we obtain
The LaplaceAdomian decomposition algorithm assumes the solution u(x,t) can be expanded into infinite series given by
Moreover, Also the nonlinear operator N is decomposed as
Each A_{n} is an Adomian polynomial of u_{0},u_{1},…,u_{n} that can be calculated for all forms of nonlinearity according to the following formula [25–27]:
Therefore Adomian’s polynomials are given by A_{0}=N(u_{0})
A_{1}=u_{1}N^{′}(u_{0})
\(A_{2}=u_{2}N'(u_{0})+\frac {1}{2}u_{1}^{2}N''(u_{0})\)
\(A_{3}=u_{3}N'(u_{0})+u_{1}u_{2}N''(u_{0})+\frac {1}{3!}u_{1}^{3}N^{(3)}(u_{0})\)
\(A_{4}=u_{4}N'(u_{0})+\left (\frac {1}{2}u_{2}^{2}+u_{1}u_{3}\right)N''(u_{0})+\frac {1}{2!}u_{1}^{2}u_{2}N^{(3)}(u_{0})+\frac {1}{4!}u_{1}^{4}N^{(4)}(u_{0})\)
⋮
All other polynomials are calculated in a similar way.
Substituting (17) and (18) into Eq. (16) gives rise to
Hence, Eq. (20) suggests the following iterative algorithm
Finally, after determining u_{n}’s, the Nterm truncated approximation of the solution is obtained as
From this analysis it is evident that, the Adomian decomposition method, combined with the Laplace transform requires less effort in comparison with the traditional Adomian decomposition method. This method considerably decreases the number of calculations. In addition, Adomian decomposition procedure is easily established without requiring to linearize the problem.
Solution of the perturbed FokasLenells equation by LADM
In this section, we outline the application of LADM to obtain explicit solution of Eq. (1) with the initial conditions u(x,0)=f(x), u_{x}(x,0)=g(x).
Let us consider the dimensionless form of the perturbed FokasLenells equation Eq. (1) in an operator form
where the notation N_{1}u=−iu^{2}(bu+iσu_{x}), N_{2}u=−λ(u^{2}u)_{x} and N_{3}u=−μu(u^{2})_{x} symbolize the nonlinear term, respectively. The notation Ru=−(αu_{x}+ia_{1}u_{xx}+ia_{2}u_{xt}) symbolize the linear differential operator and Lu=u_{t} simply means derivative with respect to time.
The LADM represents solution as an infinite series of components given below,
The nonlinear terms N_{1}u, N_{2}u and N_{3}u can be decomposed into infinite series of Adomian polynomials given by:
and
Here P_{n}, Q_{n} and R_{n} are the Adomian polynomials and can be calculated by the formula given by the Eq. (19), that is,
and for every n≥1 we have
The first few Adomian polynomials are given by
Then, the Adomian polynomials corresponding to the nonlinear part Nu=N_{1}u+N_{2}u+N_{3}u are
and so on for other Adomian polynomials.
By applying the Laplace transform with respect to t on both sides of the Eq. (23) and using the linearity of the Laplace transform gives:
Because of the differentiation property of Laplace transform, Eq. (31) can be written as
Thus,
By substituting (24), (25), (26) and (27) into (33), we obtain
Comparing both sides of the Eq. (34), the following relations arise:
In general, we get the following recursive algorithm
Finally, by applying inverse Laplace transformation we deduce the following recurrence formulas for each n=0,1,2,…,
Numerical simulations and graphical results
We perform numerical simulations for bright and dark optical solitions.
Application to bright optical solitions
The result and the profile of four cases are shown in Table 1 and in Figs. 1, 2, 3 and 4.
Application to dark optical solitions
The result and the profile of four cases are shown in Table 2 and in Figs. 5, 6, 7 and 8.
Conclusions
This paper successfully studied FLE in polarizationpreserving fibers by the aid of LaplaceAdomian decomposition scheme. The numerical scheme yielded bright and dark soliton solutions. The results thus appear with a complete spectrum of soliton solutions. Although singular solitons is a third form of solitons that emerge from this model, it does not provide any interest with any kind of numerical scheme. The results of the paper are truly encouraging to study the methodology further along. Later, this scheme will be applied to vector coupled FLE that studies solitons in birefringent fibers. Further along the model will be extended to address WDM/DWDM/UDWDM topology numerically. Such studies are currently under way.
Availability of data and materials
Not applicable.
Abbreviations
 DWDM:

Dense wavelength division multiplexing
 FLE:

Fokaslennels equation
 GVD:

Group velocity dispersion
 IMD:

Intermodal dispersion
 LADM:

Laplaceadomian decomposition method
 ND:

Nonlinear dispersion
 STD:

Spatiotemporal dispersion
 UDWDM:

Ultradense wavelength division multiplexing
 WDM:

Wavelengthdivision multiplexing
References
 1
Biswas, A., Ekici, M., Sonmezoglu, A., Alqahtani, R. T.: Optical soliton perturbation with full nonlinearity for FokasLenells equation. Optik. 165, 29–34 (2018).
 2
Biswas, A., Yildirim, Y., Yasar, E., Zhou, Q., Mahmood, M. F., Moshokoa, S. P., Belic, M.: Optical solitons with differential group delay for coupled FokasLenells equation using two integration schemes. Optik. 165, 74–86 (2018).
 3
Biswas, A., Ekici, M., Sonmezoglu, A., Alqahtani, R. T.: Optical solitons with differential group delay for coupled Fokas–Lenells equation by extended trial function scheme. Optik. 165, 102–110 (2018).
 4
Jawad Mohamad, A. J., Biswas, A., Zhou, Q., Moshokoa, S. P., Belic M.: Optical soliton perturbation of FokasLenells equation with two integration schemes. Optik. 165, 111–116 (2018).
 5
Biswas, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Ekici, M., Zhou, Q., Moshokoa, S. P., Belic, M.: Optical soliton perturbation with FokasLenells equation using three exotic and efficient integration schemes. Optik. 165, 288–294 (2018).
 6
Biswas, A.: Chirpfree bright optical soliton perturbation with FokasLenells equation by traveling wave hypothesis and semiinverse variational principle. Optik. 170, 431–435 (2018).
 7
Aljohani, A. F., Ebaid, A., ElZahar, E. R., Ekici, M., Biswas, A.: Optical soliton perturbation with FokasLenells model by Riccati equation approach. Optik. 172, 741–745 (2018).
 8
Biswas, A., Yıldırım, Y., Yaşar, E., Zhou, Q., Moshokoa, S. P., Belic, M.: Optical soliton solutions to FokasLenells equation using some different methods. Optik. 173, 21–31 (2018).
 9
Arshed, S., Biswas, A., Zhou, Q., Moshokoa, S. P., Belic, M.: Optical solitons with polarizationmode dispersion for coupled FokasLenells equation with two forms of integration architecture. Opt. Quant. Electron. 50, 304 (2018).
 10
Bansal, A., Kara, A. H., Biswas, A., Moshokoa, S. P., Belic, M.: Optical soliton perturbation, group invariants and conservation laws of perturbed FokasLenells equation. Chaos, Solitons & Fractals. 114, 275–280 (2018).
 11
Krishnan, E. V., Biswas, A., Zhou, Q., Alfiras, M.: Optical soliton perturbation with FokasLenells equation by mapping methods. Optik. 178, 104–110 (2019).
 12
Arshed, S., Biswas, A., Zhou, Q., Khan, S., Adesanya, S., Moshokoa, S. P., Belic, M.: Optical solitons pertutabation with FokasLenells equation by e x p(−ϕ(ξ))expansion method. Optik. 179, 341–345 (2019).
 13
Bansal, A., Kara, A. H., Biswas, A., Khan, S., Zhou, Q., Moshokoa, S. P.: Optical solitons and conservation laws with polarizationmode dispersion for coupled FokasLenells equation using group invariance. Chaos, Solitons & Fractals. 120, 245–249 (2019).
 14
GonzálezGaxiola, O., Biswas, A.: Wshaped optical solitons of ChenLeeLiu equation by LaplaceAdomian decomposition method. Opt. Quant. Electron. 50, 314 (2018).
 15
GonzálezGaxiola, O., Biswas, A.: Akhmediev breathers, Peregrine solitons and KuznetsovMa solitons in optical fibers and PCF by LaplaceAdomian decomposition method. Optik. 172, 930–939 (2018).
 16
GonzálezGaxiola, O., Biswas, A.: Optical solitons with RadhakrishnanKunduLakshmanan equation by LaplaceAdomian decomposition method. Optik. 179, 434–442 (2019).
 17
Fokas, A. S.: On a class of physically important integrable equations. Physica. D. 87, 145–150 (1995).
 18
Lenells, J.: Exactly solvable model for nonlinear pulse propagation in optical fibers. Stud. Appl. Math. 123, 215–232 (2009).
 19
Lenells, J., Fokas, A. S.: On a novel integrable generalization of the nonlinear Schrödinger equation. Nonlinearity. 22, 11–27 (2009).
 20
Yang, C., Liu, W., Zhou, Q., Mihalache, D., Malomed, B. A.: Onesoliton shaping and twosoliton interaction in the fifthorder variablecoefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019).
 21
Triki, H., Zhou, Q., Liu, W.: Wshaped solitons in inhomogeneous cigarshaped BoseEinstein condensates with repulsive interatomic interactions. Laser Phys. 29, 055401 (2019).
 22
Yang, C., Wazwaz, A. M., Zhou, Q., Liu, W.: Transformation of soliton states for a (2+1) dimensional fourthorder nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Laser Phys. 29, 035401 (2019).
 23
Aouadi, S., Bouzida, A., Daoui, A. K., Triki, H., Zhou, Q., Sha, Liu.: Wshaped, bright and dark solitons of BiswasArshed equation. Optik. 182, 227–232 (2019).
 24
Zhang, Y., Yang, C., Yu, W., Mirzazadeh, M., Zhou, Q., Liu, W.: Interactions of vector antidark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers. Nonlinear Dyn. 94, 1351–1360 (2018).
 25
Wazwaz, A. M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. 111(1), 33–51 (2000).
 26
Duan, J. S.: Convenient analytic recurrence algorithms for the Adomian polynomials. Appl. Math. Comput. 217, 6337–6348 (2011).
 27
Duan, J. S.: New recurrence algorithms for the nonclassic Adomian polynomials. Appl. Math. Comput. 62, 2961–2977 (2011).
Acknowledgments
Not applicable.
Funding
The research work of the third author (MRB) was supported by the grant NPRP 80281001 from QNRF and he is thankful for it.
Author information
Affiliations
Contributions
The original ideas and results emerged from discussions among all the authors. OGG wrote the manuscript with input from all authors. All authors read and approved the final manuscript.
Corresponding author
Correspondence to O. GonzálezGaxiola.
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 Fokaslenells equation
 Polarizationpreserving fibers
 Adomian decomposition method
 Optical solitons solutions
 Perturbation