Cubic-quartic optical solitons in Bragg gratings fibers for NLSE having parabolic non-local law nonlinearity using two integration schemes

The optical solitons in Bragg gratings fibers are studied for NLSE having cubic-quartic dispersive reflectivity with parabolic non-local combo law of refractive index. The extended auxiliary equation method and the addendum to Kudryashov’s method are introduced. The existence criteria for such solitons are indicated.


Introduction
In the past few decades, the nonlinear optics is one of the most effective fields of research which study the dynamics of solitons in fiber gratings. Solitons in fiber Bragg gratings (FBG) is one of the many subjects that is studied in nonlinear optics. Recently, many papers have been studied Bragg gratings models by using a lot of numerical methods (Kudryashov 2012(Kudryashov , 2020a(Kudryashov , b, 2021aBiswas et al. 2019a, b, c;Sonmezoglu et al. 2016;Xu 2014;Zayed et al. 2020a, b, c;Bansal et al. 2018;Biswas et al. 2017a, b;Yıldırım et al. 2020a, b;Zayed et al. 2021a, b;Yıldırım et al. 2021a, b, c, d;Zayed et al. 2021cZayed et al. , 2020dYıldırım et al. 2021e;Darwish et al. 2020;Xu et al. 2020;Chen et al. 2020;Xia et al. 2020;Chen et al. 2021a, b;. There are several approaches to handle this project. A great amount of results have been introduced in these papers. In such a situation, Bragg gratings artificially introduces induced dispersion that restores this balance for sustainability of soliton transmission along intercontinental distances. Newly, in the field of nonlinear fiber optics, the concept of cubic-quartic (CQ) optical solitons has been introduced in many papers (Bansal et al. 2018;Biswas et al. 2017a, b;Yıldırım et al. 2020a, b;Zayed et al. 2021a, b;Yıldırım et al. 2021a, b, c, d;Zayed et al. 2021c). When the chromatic dispersion (CD) runs low enough to be

Governing model
The coupled CQ-NLSE in Bragg gratings fibers with parabolic non-local law nonlinearity is written as: and where a l , b l , c l , d l , l , l , l , f l , g l , l , l and l , (l = 1, 2) are real parameters such that i = √ −1 . Here, u(x, t) , v(x, t) are the complex wave profiles. The coefficients of a l , b l are 3OD and 4OD, respectively. The parameters c l , l , f l represent the self-phase modulation (SPM) coefficients, while the cross-phase modulation (XPM) effect comes from the coefficients d l , l , l and g l . The parameters l , l and l are the coefficients of inter-modal dispersion (IMD), detuning parameter and four-wave mixing effect (4WM) for Kerr part of the nonlinearity, respectively. The system (1) and (2) is a manifested version of the standard model. That is the well-known NLSE model in FBG with CD, that is structured as Zayed et al. (2020d): and where E 1 and E 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. The objective of this paper is to apply the extended auxiliary equation method and the addendum to Kudryashov's method to find the bright, dark and singular solitons solutions as well as the Jacobi elliptic function solutions of the coupled system (1) and (2).
The organization of this article can be written as: the mathematical preliminaries are discussed in Sect. 2. The extended auxiliary equation method is applied to the coupled system (1) and (2) in Sect. 3. The addendum to Kudryashov's method is applied to the same coupled system in Sect. 4. Lastly, conclusions are given in Sect. 5.

Mathematical preliminaries
In order to recover solitons of the CQ-NLSE in fiber Bragg gratings with parabolic nonlocal law nonlinearity, we set and where C, , and 0 are all non zero parameters. Here, C is the velocity of soliton, is the frequency of soliton, is the wave number of the soliton and finally, 0 is the phase parameter, while P 1 ( ), P 2 ( ) and (x, t) are real functions representing the amplitude portion of the soliton and the phase component of the soliton, respectively. If we substitute (5) and (6) into Eqs. (1) and (2) and separate the real and imaginary parts, we deduce that the real parts are and the imaginary parts are

The extended auxiliary equation method
According to this method (Xu 2014;Zayed et al. 2020a, b, c), we assume that Eq. (31) has the formal solution where A 0 , A 1 and A 2 are constants to be determined, such that A 2 ≠ 0, while the function F( ) satisfies the following first order equation: where C j (j = 0, 2, 4, 6) are constants to be determined. It is well known that Eq. (34) has the following solution: where f ( ) could be expressed through the Jacobi elliptic functions sn ( , m), cn ( , m), dn ( , m) and so on. Here 0 < m < 1 is the modulus of the Jacobi elliptic functions. Substituting (33) along with (34) into Eq. (31), collecting the coefficients of each power F l ( ) F � ( ) j , (l = 0, 1, 2, .., 10 , j = 0, 1) and setting these coefficients to zero, we have a set of algebraic equations which can be solved using the Maple to obtain the following results: provided L 5 > 0. From (33), (35) and (36), then we have the solutions: We have the following families of Jacobi elliptic functions solutions of Eqs. (1) and (2): , C 2 = C 2 4 5m 2 − 1 16C 6 m 2 , C 6 > 0 , then and or (33) and provided L 5 > 0 (Fig. 1).