Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 19, Number 2, 2023



Article Number  38  
Number of page(s)  12  
DOI  https://doi.org/10.1051/jeos/2023035  
Published online  25 July 2023 
Research Article
Dynamical system of optical soliton parameters by variational principle (superGaussian and supersech pulses)
^{1}
Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
^{2}
Basic Science Department, Faculty of Computers and Artificial Intelligence, Modern University for Technology & Information, Cairo 11585, Egypt
^{3}
Basic Science Department, Higher Institute of Foreign Trade & Management Sciences, New Cairo Academy, Cairo 379, Egypt
^{4}
Department of Mathematics and Physics, Grambling State University, Grambling, LA 71245, USA
^{5}
Mathematical Modeling and Applied Computation (MMAC) Research Group, Center of Modern Mathematical Sciences and their Applications (CMMSA), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
^{6}
Department of Applied Sciences, CrossBorder Faculty of Humanities, Economics and Engineering, Dunarea de Jos University of Galati, 111 Domneasca Street, Galati 800201, Romania
^{7}
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, South Africa
^{8}
Department of Computer Engineering, Biruni University, 34010 Istanbul, Turkey
^{9}
Department of Mathematics, Near East University, 99138 Nicosia, Cyprus
^{10}
Faculty of Sciences and Environment, Department of Chemistry, Physics and Environment, Dunarea de Jos University of Galati, 47 Domneasca Street, 800008, Romania
^{11}
Department of Business Administration, Faculty of Economics and Business Administration, Dunarea de Jos University of Galati, 59–61 Nicolae Balcescu Street, 800001 Galati, Romania
^{*} Corresponding author: luminita.moraru@ugal.ro
Received:
3
June
2023
Accepted:
6
July
2023
The parameter dynamics of supersech and superGaussian pulses for the perturbed nonlinear Schrödinger’s equation with powerlaw nonlinearity is obtained in this article. The variational principle successfully recovers this dynamical system. The details of the variational principle with the implementation of the Euler–Lagrange’s equation to the nonlinear Schrödinger’s equation with powerlaw of nonlinearity described in this paper have not been previously reported.
Key words: Solitons / Variational principle / Perturbation / Euler–Lagrange
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The dynamics of optical solitons is a long standing study that has now extended over halfacentury. Various aspects of soliton science have been reported. Notably, most of the papers are from the integrability aspects of a variety of models that arose from wide range of selfphase modulation (SPM) structures. A few papers are from additional, sparingly visible, topics such as conservation laws, quasimonochromatic solitons with the usage of perturbation theory, stochastic perturbation and the corresponding mean free velocity of the soliton and others.
One of the most viable topics that serves as an important foundation stone in optical soliton dynamics is the recovery of the soliton parameter dynamics such as the amplitude, width, center position, phase constant and similar such parameters. This can be achieved in several different ways. A few such mathematical approaches are the soliton perturbation theory, collective variables approach and the moment method. However, for example, soliton perturbation theory has its shortcomings. It fails to recover the variation of the phase constant as well as the variation of the center position of the soliton. The variational principle (VP) overcomes this hurdle. This has been succesfully and widely applied to various areas of Physics and Engineering such as Condensed Matter Physics, Fluid Dynamics and Fiber Optics including dispersionmanaged solitons [1–20].
The advantages and necessity of obtaining the dynamical system of soliton parameters are multifold. The study of soliton features in optics can be further enhanced through the utilization of these parameter dynamics. Four wave mixing effects, collisioninduced timing jitter, and various other phenomena are among those that are included. Therefore, the parameter dynamics with the existence of perturbation terms is being studied by applying the VP to the nonlinear Schrödinger’s equation (NLSE). Supersech and superGaussian pulses are the two types of pulses being examined in this context. This would give a generalized flavor to the study of soliton parameters. The details of the VP with the implementation of the Euler–Lagrange’s equation to NLSE with powerlaw of nonlinearity described in this paper have not been previously reported. A quick and succinct introduction is followed by the presentation of results.
2 Unperturbed NLSE with powerlaw nonlinearity
The governing model of such equation is written as:(1)where the coefficients b and a are utilized to denote SPM and chromatic dispersion in sequence. The function q = q(x, t) represents the wave profile in a complexvalued form, where . Equation (1) contains the linear temporal evolution, represented by the first term.
2.1 Variational principle
The Lagrangian (L) is associated with equation (1) is written as:(2)
One obtains q* by complexconjugating q. In equation (1), the assumed pulse q = q(x, t) is presented as:(3)
We use the symbols θ_{0}(t), κ(t), , B(t), and A(t) to denote the soliton phase, soliton frequency, center position of the soliton, pulse width, and soliton amplitude, respectively. Setting(4)then the pulse hypothesis (3) becomes(5)
Through the application of the provided equation(6)we conclude that:(7)and(8)
Substituting (5)–(8) into (2) and using the formula ds = B(t)dx the Lagrangian (2) reduces to(9)where(10)and nonnegative integers are the only values that a, b, and c can assume.
The integrals of motion can be obtained from the pulse form (5), which can be derived, as presented below(11) (12)
The mathematical representation of the Hamiltonian takes the form of(13)
2.2 Parameter dynamics of the NLSE
Introducing the following EulerLagrange (EL) equation [4, 8] in this subsection leads to the derivation of the dynamical system:(14)where the soliton parameters A(t), B(t), , and are represented by the variable p, where p denotes one of them. The dynamic system below is derived by substituting (9) into (14):(15) (16) (17) (18)and(19)
For the pulse form given by (5), the equations (15)–(19) provide the general forms of the soliton parameter dynamics of equation (1). The dynamic system (15)–(19) can be expressed in a simplified and reduced form as:(20) (21) (22) (23)and(24)where the square roots of the energy are proportional to the constant K in (24). From (21) and (24), we have:(25)
2.3 SuperGaussian pulses
Assuming m > 0, the super Gaussian pulse function can be written as . Then, one can obtain the integrals of motion as:(26) (27)and the Hamiltonian is given by:(28)
For u > 0, the gamma function is defined as Γ(u). This compels the parameter m to be bounded below as given by(29)
The pulse parameters can be obtained from the evolution equations (20)–(25), which can be expressed in a reduced form as:(30) (31) (32) (33) (34)and(35)
Figures 1 and 2 provide a few plots of superGaussian pulse and supersech pulse with the governing model (1), respectively. These plots offer a visual depiction of the waveform characteristics and provide valuable insights into the behavior of the pulses under investigation. The parameter vales chosen are: K = 1, κ(t) = 1, a = 1, , b = 1, n = 1,5 and m = 2,5.
Figure 1 Profile of a superGaussian pulse. (a) Surface plot. (b) 2D plots moving in time. (c) Contour plot. 
Figure 2 Profile of a supersech pulse. (a) Surface plot. (b) 2D plots moving in time. (c) Contour plot. 
2.4 Supersech pulses
For supersech pulses, we set f(s) = sech^{2m s}, m > 0 Then, one can address the equations governing the integrals of motion that are expressed as:(36) (37)and we can express the Hamiltonian as:(38)
Here, the generalized form of Gauss’ hypergeometric function is expressed as:(39)and the symbol for the Pochhammer is:(40)
The pulse parameters are governed by the evolution equations (20)–(25), which can be expressed in a simpler form as:(41) (42) (43) (44) (45)and(46)
3 Perturbed NLSE with powerlaw nonlinearity
The equation is described by the following governing model:(47)where is given by:(48)and ϵ, δ, α, β, λ, θ, σ, ξ, η, ζ, μ, σ_{1} and σ_{2} are constants, where ϵ is from quasimonochromaticity. From (5) and (47), we have(49)
3.1 Parameter dynamics of the perturbed NLSE
In this subsection, we derive the dynamical system of equation (47) by introducing the following Euler Lagrange (EL) equation:(50)where L is given by (9) and p is one of these same five parameters A(t), B(t), , κ(t) and θ_{0}(t), respectively, while R^{*} is the complexconjugate of R. Now, we have the following dynamic system:(51) (52) (53) (54)and(55)
The general forms of the soliton parameters dynamics of equation (47) for the pulse form given by (5) are represented by equations (51)–(55). A simplified version of the dynamic system (51)–(55) is:(56) (57) (58) (59) (60)where A = A(t), B = B(t) and κ = κ(t).
3.2 SuperGaussian pulses
The dynamical system (56)–(60) is reduced to a simpler form for superGaussian pulses, which is:(61) (62) (63) (64) (65)
The equations involve the incomplete gamma function, which is represented by Γ(a, x).
3.3 Supersech pulses
The dynamical system (56)–(60) simplifies to a specific form when considering supersech pulses, as described below(66) (67) (68) (69) (70)
4 Conclusions
Our study recovers the dynamical system of soliton parameters for supersech and superGaussian pulses, as described in this paper. The details of the VP with the implementation of the Euler–Lagrange’s equation to the NLSE with powerlaw of nonlinearity indicated in the current work have not been previously reported. These parameter variations, namely the dynamical system opens up with an avalanche of opportunities to study optical soliton sciences further along. This foundation stone of results pave way to further future investigations in this chapter. Later, the dynamical system would be revealed for additional forms of SPM that have not yet been considered. The studies would later be extended to birefringent fibers and DWDM topology. These would give an increased perspective to carry out the analysis further along. This would also be applicable to various additional devices and other forms of waveguides, including optical metamaterials, magnetooptic waveguides, optical couplers, gap solitons and many others. The results of these studies will be reported soon after we align them with the preexisting ones [21–25]. All of these activities are currently underway.
References
 Ali S.G., Talukdar B., Roy S.K. (2007) Bright solitons in asymmetrically trapped BoseEinstein condensates, Acta Phys. Pol. A 111, 3, 289–297. [NASA ADS] [CrossRef] [Google Scholar]
 Ayela A.M., Edah G., Elloh C., Biswas A., Ekici M., Alzahrani A.K., Belic M.R. (2021) Chirped superGaussian and supersech pulse perturbation of nonlinear Schrödinger’s equation with quadratic–cubic nonlinearity by variational principle, Phys. Lett. A 396, 127231. [NASA ADS] [CrossRef] [Google Scholar]
 Ayela A.M., Edah G., Biswas A., Zhou Q., Yildirim Y., Khan S., Alzahrani A.K., Belic M.R. (2022) Dynamical system of optical soliton parameters for anti–cubic and generalized anti–cubic nonlinearities with super–Gaussian and super–sech pulses, Opt. Appl. 52, 1, 117–128. [Google Scholar]
 Biswas A. (2001) Dispersion–managed solitons in optical fibres, J. Opt A Pure Appl. Op. 4, 1, 84–97. [Google Scholar]
 Chen Y. (1991) Variational principle for vector spatial solitons and nonlinear modes, Opt. Commun. 84, 5–6, 355–358. [NASA ADS] [CrossRef] [Google Scholar]
 Diakonos F.K., Schmelcher P. (2019) SuperLagrangian and variational principle for generalized continuity equations, J. Phys. A. 52, 155–203. [Google Scholar]
 Ferreira M.F.S. (2018) Variational approach to stationary and pulsating dissipative optical solitons, IET Optoelectron. 12, 3, 122–125. [CrossRef] [Google Scholar]
 Green P., Milovic D., Sarma A.K., Lott D.A., Biswas A. (2010) Dynamics of super–sech solitons in optical fibers, J. Nonlinear Opt. Phys. Mater. 19, 2, 339–370. [NASA ADS] [CrossRef] [Google Scholar]
 Hirooka T., Wabnitz S. (2000) Nonlinear gain control of dispersion–managed soliton amplitude and collisions, Opt. Fiber Technol. 6, 2, 109–121. [NASA ADS] [CrossRef] [Google Scholar]
 Latas S., Ferreira M. (2010) Soliton explosion control by higher–order effects, Opt. Lett. 35, 1771–1773. [NASA ADS] [CrossRef] [Google Scholar]
 Mancas S., Choudhury S. (2007) A novel variational approach to pulsating solitons in the cubic–quintic GinzburgLanadu equation, Theor. Math. Phys. 152, 1160–1172. [NASA ADS] [CrossRef] [Google Scholar]
 Pal D., Ali S.G., Talukdar B. (2008) Embedded soliton solutions: A variational study, Acta Phys. Pol. A. 113, 2, 707–712. [NASA ADS] [CrossRef] [Google Scholar]
 Rubinstein J., Wolansky G. (2004) A variational principle in optics, J. Opt. Soc. Am. B. 21, 11, 2164–2172. [NASA ADS] [CrossRef] [Google Scholar]
 Skarka V., Aleksic N.B. (2007) Dissipative optical solitons, Acta Phys. Pol. A 112, 5, 791–798. [NASA ADS] [CrossRef] [Google Scholar]
 Zhang J., Yu J.Y., Pan N. (2005) Variational principles for nonlinear fiber optics, Chaos Solit. Fractals 4, 309–311. [CrossRef] [Google Scholar]
 Zhou Q. (2022) Influence of parameters of optical fibers on optical soliton interactions, Chin. Phys. Lett. 39, 1, 010501. [NASA ADS] [CrossRef] [Google Scholar]
 Ding C.C., Zhou Q., Triki H., Hu Z.H. (2022) Interaction dynamics of optical dark bound solitons for a defocusing LakshmananPorsezianDaniel equation, Opt. Exp. 30, 22, 40712–40727. [NASA ADS] [CrossRef] [Google Scholar]
 Wang H., Zhou Q., Liu W. (2022) Exact analysis and elastic interaction of multisoliton for a twodimensional GrossPitaevskii equation in the BoseEinstein condensation, J. Adv. Res. 38, 179–190. [CrossRef] [Google Scholar]
 Feng W., Chen L., Ma G., Zhou Q. (2022) Study on weakening optical soliton interaction in nonlinear optics, Nonlinear Dyn. 108, 3, 2483–2488. [CrossRef] [Google Scholar]
 Wang T.Y., Zhou Q., Liu W.J. (2022) Soliton fusion and fission for the highorder coupled nonlinear Schrödinger system in fiber lasers, Chin. Phys. B 31, 2, 020501. [NASA ADS] [CrossRef] [Google Scholar]
 Zhou Q., Huang Z., Sun Y., Triki H., Liu W., Biswas A. (2023) Collision dynamics of threesolitons in an optical communication system with thirdorder dispersion and nonlinearity, Nonlinear Dyn. 111, 6, 5757–5765. [CrossRef] [Google Scholar]
 Ding C.C., Zhou Q., Triki H., Sun Y., Biswas A. (2023) Dynamics of dark and antidark solitons for the xnonlocal DaveyStewartson II equation, Nonlinear Dyn. 111, 3, 2621–2629. [CrossRef] [Google Scholar]
 Zhong Y., Triki H., Zhou Q. (2022) Analytical and numerical study of chirped optical solitons in a spatially inhomogeneous polynomial law fiber with paritytime symmetry potential, Commun. Theoret. Phys. 75, 025003. [Google Scholar]
 Zhou Q., Triki H., Xu J., Zeng Z., Liu W., Biswas A. (2022) Perturbation of chirped localized waves in a dualpower law nonlinear medium, Chaos, Solitons & Fractals 160, 112198. [NASA ADS] [CrossRef] [Google Scholar]
 Zhou Q., Zhong Y., Triki H., Sun Y., Xu S., Liu W., Biswas A. (2022) Chirped bright and kink solitons in nonlinear optical fibers with weak nonlocality and cubicquanticseptic nonlinearity, Chin. Phys. Lett. 39, 4, 044202. [NASA ADS] [CrossRef] [Google Scholar]
All Figures
Figure 1 Profile of a superGaussian pulse. (a) Surface plot. (b) 2D plots moving in time. (c) Contour plot. 

In the text 
Figure 2 Profile of a supersech pulse. (a) Surface plot. (b) 2D plots moving in time. (c) Contour plot. 

In the text 
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