Optical solitons in ﬁber Bragg gratings with quadratic-cubic law of nonlinear refractive index and cubic-quartic dispersive reﬂectivity

. This paper recovers cubic-quartic perturbed solitons in ﬁber Bragg gratings with quadratic-cubic law nonlinear refractive index. The uniﬁed Riccati equation expansion method and the modiﬁed Kudryashov’s approach make this retrieval of soliton solutions possible. The parameter constraints, for the existence of such solitons, are also presented


INTRODUCTION
The theory of solitons in fiber Bragg is one of the essential areas of research in nonlinear photonics and it has been studied both analytically and numerically [1][2][3][4][5][6][7][8][9][10][11][12]. The concept of fiber Bragg gratings (FBG) is a couple of decades old; however, the research on this topic remains active, but with an advanced and enhanced perspective. The concept of fiber Bragg gratings emerged when chromatic dispersion (CD) in soliton transmission dynamics got negligibly small and it is then that Bragg gratings were introduced in an optical fiber to compensate for the low-lying balance between CD and self-phase modulation (SPM). While this is a modern engineering marvel in the technology of optical fiber materials, another approach is to discard CD due to its low count and replace it collectively with third-order dispersive (3OD) and fourth-order dispersive (4OD) effects. This is known as cubic-quartic (CQ) dispersive effect. This paper will address optical fibers with fiber Bragg gratings with dispersive reflectivity from CQ dispersive effect [13][14][15][16][17][18][19][20]. The nonlinear law of refractive index will be sourced from quadratic-cubic (QC) terms. This is the model that will be analyzed in the paper using two integration algorithms which lead to a full spectrum of single soliton solutions that will be exhibited. The existence criteria of such solutions are also presented. The rest of the paper pens the derivation and displays these soliton solutions after a quick and succinct intro to the governing model.

Governing Model
The dimensionless form of the coupled cubic-quartic (CQ) perturbed nonlinear Schrödinger equation (NLSE) in fiber Bragg gratings (FBG) with quadratic-cubic law of nonlinear refractive index, is written as where a j and b j ( j = 1, 2) are the coefficients of 3OD and 4OD, respectively. Here q (x,t) and r (x,t) represent forward and backward propagation wave profile, respectively, as long as i = √ −1. Then, d j are the coefficients of SPM terms; c j and e j represent the cross-phase modulation (XPM) terms; α j represent the inter-modal dispersion and β j are the detuning parameters. Also, σ j give the four wave-mixing (4WM) terms, while γ j represent the self-steepening terms to avoid the shock waves formulation. Finally, θ j and µ j read as the nonlinear dispersion coefficients.
The system (1) and (2) is a manifested version of the standard model, which comes with CD instead of CQ dispersive effect, and is structured as iq t + a 1 r xx + c 1 q |q| 2 + |r| 2 + q * r + qr * + d 1 |q| 2 + e 1 |r| 2 q ir t + a 2 q xx + c 2 r |r| 2 + |q| 2 + r * q + rq * + d 2 |r| 2 + e 2 |q| 2 r where a j ( j = 1, 2) are the coefficients of CD. In the system (1) and (2), it is this CD that is replaced by 3OD and 4OD which formulate the dispersion effects. Note that, the system (3) and (4) has been discussed in [18], when γ j = θ j = µ j = 0, for j = 1, 2. The objective of this paper is to apply two algorithms techniques mentioned in the abstract to construct the optical soliton solutions of the CQ-system (1) and (2).
This article is organized as follows: Section 2 introduces the mathematical preliminaries. In Sections 3 and 4, the unified Riccati equation expansion method and the modified Kudryashov's method are applied. In Section 5, conclusions are drawn.

MATHEMATICAL PRELIMINARIES
In order to solve Eqs (1) and (2), we assume the hypothesis such that where v, κ, ω and θ 0 are all nonzero constants to be determined, which represent soliton velocity, soliton frequency, wave number and phase constant, respectively. Next, ψ(x,t) is a real function which represents the phase component of the soliton, while φ j (ξ ) are real functions which represent the shape of the pulse of the solitons. Substituting (5) and (6) into Eqs (1) and (2), separating the real and imaginary parts, we deduce that the real parts are while the imaginary parts are Set where A = 0 or 1. Consequently, the real parts change into while the imaginary parts become The linearly independent principle is applied on (14) and (15) to get the frequency of soliton as the velocity of soliton as and the constraint conditions: From (17) and (18), one gets the following constraint condition: Eqs (12) and (13) have the same form under the constraint conditions: From (21), we have Eq. (12) can be rewritten as where Next, Eq. (23) will be examined.

APPLICATION OF MODIFIED KUDRYASHOV'S METHOD
There are several approaches and results of Kudryashov that have gained a considerable popularity lately due to its unique integrability structure [21][22][23][24][25][26][27][28][29][30]. Modified Kudryashov's approach proposes the formal solution to Eq. (23) as given below where ρ j are constants to be determined later, provided ρ N = 0 and Q(ξ ) takes the form with the positive integer p, such that the solutions of Eq. (46) are given by where exp a (pξ + ξ 0 ) = a pξ +ξ 0 , ξ 0 is a constant and ε = ±1. Eq. (47) can be rewritten in combination with bright-singular soliton solution as In particular if ε = 1, one gets the dark soliton solution When ε = −1, the singular soliton solution is obtained Balancing φ 1 with φ 3 1 in Eq. (23), one gets the relation Next, two cases will be studied.
Similarly, one can derive many other solutions by selecting other values for p and N.

CONCLUSIONS
This paper recovered single optical soliton solutions in fiber Bragg gratings that come with CQ dispersive reflectivity with QC nonlinear law of refractive index. The model includes a few Hamiltonian type perturbation terms. Two integration algorithms have made this retrieval possible. Single soliton solutions of all kinds are recovered and are listed with their respective parameter constraints. The results of this form of a model are being reported for the first time and a lot of work lies ahead of us with such a display. One of the immediate topics to address is the conservation laws that would make a big difference as this essential feature is noticeably missing in the literature of Bragg gratings. Therefore, it is imperative to bridge this gap. Many other features such as addressing this model with fractional temporal evolution or time-dependent coefficients or even handling the study of solitons, with Bragg gratings, using Lie symmetry, are all yet to be done along the lines of the previously reported studies [31][32][33][34][35][36][37][38][39][40]. These ambitious projects still form only the tip of the iceberg! One must also make a cautious approach and try to avoid reporting solutions that will later be identified as false [31].