Optical soliton polarization with Lakshmanan–Porsezian–Daniel model by unified approach

Introduction Optical soliton dynamics is an engineering marvel in telecommunications industry [1–10]. An inherent problem with the dynamics of pulse propagation across trans-oceanic and trans-continental distances is its polarization. This is attributed to several factors such as the randomness of fiber diameter, rough handling of optical fibers and many others. These factors occasionally lead to hi-bi fibers. It is often a challenging task to retrieve the soliton solutions to the models that are studied in the context of high birefringence. One such model that has been around for a fairly long period is the Lakshmanan–Porsezian–Daniel (LPD) model that was first reported in 1988 and later gained a lot of popularity [11]. A wide range of integration algorithms have been implemented to secure soliton and other solutions to LPD model in the context of polarization-preserving fibers [12], including exp(−φ(ξ))-expansion scheme [13], trial equation scheme [14] and many more [15–18]. Today’s work will retrieve ∗ Corresponding author. soliton solutions to LPD model with differential group delay by unified approach that was first reported during 2018 [19]. As it will be revealed, the algorithm could only expose singular solitons. The details are jotted in the rest of the paper after a quick re-visitation of the model and the integration algorithm. Governing model The dimensionless LPD model with Kerr law nonlinearity has the following form [15,16,20]: iψt + aψxx + bψxt + c |ψ| 2 ψ = σψxxxx + pψ2 xψ ∗ + q | | ψx| 2 ψ + r |ψ|2 ψxx + λψ2ψ∗ xx + s |ψ| 4 ψ. (1) In Eq. (1), x and t represent independent spatial and temporal variables, respectively. The dependent variable ψ represents the complex wave vailable online 11 February 2021 211-3797/© 2021 The Authors. Published by Elsevier B.V. This is an open access a E-mail address: mehmet.ekici@bozok.edu.tr (M. Ekici). https://doi.org/10.1016/j.rinp.2021.103958 Received 20 November 2020; Received in revised form 4 February 2021; Accepted rticle under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 5 February 2021 Results in Physics 22 (2021) 103958 M.S. Ullah et al.


Introduction
Optical soliton dynamics is an engineering marvel in telecommunications industry [1][2][3][4][5][6][7][8][9][10]. An inherent problem with the dynamics of pulse propagation across trans-oceanic and trans-continental distances is its polarization. This is attributed to several factors such as the randomness of fiber diameter, rough handling of optical fibers and many others. These factors occasionally lead to hi-bi fibers. It is often a challenging task to retrieve the soliton solutions to the models that are studied in the context of high birefringence.
One such model that has been around for a fairly long period is the Lakshmanan-Porsezian-Daniel (LPD) model that was first reported in 1988 and later gained a lot of popularity [11]. A wide range of integration algorithms have been implemented to secure soliton and other solutions to LPD model in the context of polarization-preserving fibers [12], including exp(− ( ))-expansion scheme [13], trial equation scheme [14] and many more [15][16][17][18]. Today's work will retrieve * Corresponding author.
soliton solutions to LPD model with differential group delay by unified approach that was first reported during 2018 [19]. As it will be revealed, the algorithm could only expose singular solitons. The details are jotted in the rest of the paper after a quick re-visitation of the model and the integration algorithm.

Governing model
The dimensionless LPD model with Kerr law nonlinearity has the following form [15,16,20]: function. Next, the parameters , , , and signify group velocity dispersion, spatio-temporal dispersion, the coefficient of Kerr law nonlinearity, the coefficient of fourth order dispersion and the two-photon absorption, respectively. Finally, the , , and terms account for several forms of the nonlinear dispersion. Solitons are possible for a sustained delicate balance of dispersion with the nonlinear terms. For birefringent fibers, the model can be divided into two parts of a vector representation. Avoiding the properties of 4WM, the above model reduces to [13,14]: In Eqs. (2) and (3), , for = 1, 2 represent the self-phase and , , with = 1, 2 stand for the cross-phase modulation effects, respectively.

Mathematical analysis
Consider the following transformation of this coupled system where 1 and 2 are the soliton amplitude components and is the traveling wave variable with the soliton speed . The phase component is as below: with frequency , wave number and phase shift . Inserting Eqs. (4) and (5) into Eqs. (2) and (3) and sorting out the real and imaginary parts leads to the following equations. The real part is while the imaginary part is with = 1, 2 and̄= 3 − . By the balancing principle, one can writē From Eqs. (8) and (10), we can rewrite From Eqs. (9) and (10), one can rewrite Thus, the third expression of Eq. (13) gives = 0. Hence the solutions of the coupled system Eqs. (2) and (3) will be presented for the fourth order dispersion omitted. The other terms in Eq. (13), yield the following relation and therefore the soliton speed is Comparing the values of the soliton velocity, Eq. (15) gives Therefore Eq. (11) can be written as

Case-3:
Rational function solution (when = 0): To identify the value of in Eq. (18), balancing 2 ′′ with 5 yields = 1. Eq. (18) takes the form which are all singular soliton pairs. Further, one arrives at the following periodic solutions: These constitute another set of singular solitons. Next, the periodic solution pairs are

Conclusions
This paper revealed soliton solutions to LPD model with differential group delay. The polarized solitons are thus retrieved and exhibited. The scheme implemented is the unified approach which yielded singular soliton solutions only. Singular solitons are applicable to model optical rogons, but not optical solitons, and the algorithm, evidently, has a few drawbacks. The method fails to retrieve the much-needed bright solitons and dark solitons. Also, this scheme is unable to produce -soliton solutions to the governing model. Moreover, a profound drawback is its inability to locate soliton radiation that is inevitably present once linear and nonlinear dispersion terms are embedded in the model. Thus, to conclude, the unified approach is not of much use in the study of governing models that give rise to optical solitons. From the applications perspective, this integration algorithm cannot be applied to obtain bright or dark solitons in any model. It can only be used to address optical rogons that are supposedly modeled with singular solitons. In addition, singular solitons cannot be plotted. This paper therefore concludes with analytical results for only singular solitons obtained by the aid of the unified method.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.