Highly Dispersive Optical Solitons with Complex Ginzburg–Landau Equation Having Six Nonlinear Forms

: This paper retrieves highly dispersive optical solitons to complex Ginzburg–Landau equation having six forms of nonlinear refractive index structures for the very ﬁrst time. The enhanced version of the Kudryashov approach is the adopted integration tool. Thus, bright and singular soliton solutions emerge from the scheme that are exhibited with their respective parameter constraints.


Introduction
The physics and technology of optical solitons in telecommunications industry has totally revolutionized the modern world of quantum communications. The dynamics of soliton propagation through a variety of waveguides [1][2][3], as well as the modern study of meta-optics covers it all. Later, the concept of highly dispersive (HD) optical solitons [4][5][6][7][8] that was conceived during 2019 has theoretically addressed a growing problem in the modern telecommunications industry. This is the low count of chromatic dispersion (CD) that is a key element in sustaining the much needed balance between it and the self-phase modulation (SPM). HD solitons provide additional sources of dispersion to maintain this key balance between CD and SPM for the smooth propel of solitons through optical fibers for trans-continential and trans-oceanic distances. These additional sources of dispersion are from inter-modal dispersion (IMD), third-order dispersion (3OD), fourthorder dispersion (4OD), fifth-order dispersion (5OD), and sixth-order dispersion (6OD). These lead to the concept of HD solitons although, technically, dispersive effects would dominate the soliton propagation. Another shortcoming would be the drastic slow-down of solitons with such a collective dispersive count.
When HD solitons first came into existence, it was on the platform of nonlinear Schrödinger's equation (NLSE) [9][10][11][12]. After the concept of HD solitons was first reported, several works from this area have flooded a variety of journals over the last couple of years [13][14][15][16][17]. This, in fact, includes addressing of solitons with eighth-order dispersion.
The current paper is addressing, for the first time, HD solitons on a different platform, namely the complex Ginzburg-Landau equation (CGLE) [18][19][20][21][22][23][24][25][26]. There are six forms of nonlinear refractive index structures that are considered. The integration scheme is the enhanced Kudryashov approach that reveals bright and singular optical solitons for each of these six nonlinear forms. These are exhibited and their respective parameter constraint conditions are also displayed. The detailed analysis are pen-pictured after a quick intro to the model.

Governing Model
The perturbed HD-CGLE that is considered for the very first time in this paper is indicated below iq t + ia 1 q x + a 2 q xx + ia 3 q xxx + a 4 q xxxx + ia 5 q xxxxx + a 6 q xxxxxx + F |q| 2 q = α |q x | 2 q * + β 4|q| 2 q * 2|q| 2 |q| 2 xx − |q| 2 where q = q(x, t) denotes the wave profile and q * represents the complex conjugate of the field q = q(x, t), while t and x represents temporal and spatial variables, sequentially. a j (j = 1, 2, · · · , 6) are the coefficients of IMD, CD, 3OD, 4OD, 5OD, and 6OD. The first term is linear temporal evolution and i = √ −1. γ gives the detuning effect. λ is the coefficient of self-steepening. µ is the coefficient of higher-order dispersion. υ is the coefficient of nonlinear dispersion. β and α are the coefficients of nonlinear term. Lastly, F |q| 2 stands for nonlinear form.
Equation (1) is a generalized version of the perturbed CGLE [27][28][29][30][31] This paper studies the perturbed HD-CGLE (1) with six nonlinear forms using the integration methodology. The current paper is structured as: In Section 2, the perturbed HD-CGLE (1) is analyzed. In Section 3, the integration methodology is presented. In Sections 4-9, we arrive soliton solutions with the proposed models. The results of the paper are discussed in Section 10.

Mathematical Preliminaries
We presume the traveling wave transformation where φ(ξ) is the amplitude of the traveling wave, κ is the frequency, c is the velocity, w is the wave number and θ 0 is the phase constant. Substituting (3) into (1) gives the real part a 6 φφ (6) + 5a 5 κ + a 4 − 15a 6 κ 2 φφ (4) and the imaginary part Equation (5) yields the velocity and the frequency and the constraint conditions Equation (4) can be written as where

Conclusions
This paper reports HD solitons with the perturbed CGLE having six forms of nonlinear refractive index structures. The perturbation terms are all of Hamiltonian type and are with maximum intensity. The integration scheme is the enhanced Kudryashov approach that is the extended and generalized version of the pre-existing Kudryashov scheme. Thus, straddled, bright and singular soliton solutions emerge from the scheme for each of these six nonlinear forms that are exhibited with their respective parameter constraints. Likewise, one can obtain a large number of HD solitons of the model equations by taking a different selection to the parameters p and N.
There is an abundance of results that have been retrieved on the perturbed CGLE [16,28,29] where Hamiltonian type perturbation terms are studied with maximum intensity. A spectrum of cubic-quartic optical solitons for the perturbed CGLE having a variety of six forms of nonlinear refractive index structures are derived by eight powerful and prolific integration structures [28]. Additionally, conservation laws for pure-cubic optical solitons with the perturbed CGLE having eleven forms of nonlinear refractive index structures are derived with the implementation of Lie symmetry analysis [29]. Lastly, pure-cubic optical solitons with the perturbed CGLE having a dozen nonlinear refractive index structures are recovered by two integration schemes [16]. However, compared with [16,28,29] that secure pure-cubic or cubic-quartic optical solitons with the model equation, HD solitons with the perturbed CGLE are given in the current paper for the very first time. The results of this work are with unprecendented novelty and thus carry tremendous value in further future development of the concept of HD solitons with CGLE and/or NLSE. The results are indeed promising.
Later, conservation laws will be reported. The variational principle would lead to the evolution of the soliton parameters in presence and/or absence of perturbative effects. Once these fundamental results are in place, one can move further along with the development of quasi-particle theory to suppress intra-channel collision of optical solitons. The quasistationarity will also be addressed to recover soliton solutions in presence of perturbation terms, be it Hamiltonian or non-Hamiltonian. These are just a few droplets of a wide and deep ocean!!  Data Availability Statement: All data generated or analyzed during this study are included in this manuscript.

Acknowledgments:
The authors thank the anonymous referees whose comments helped to improve the paper.

Conflicts of Interest:
The authors declare no conflict of interest.