Optical solitons in Bragg gratings fibers for the nonlinear (2+1)-dimensional Kundu–Mukherjee–Naskar equation using two integration schemes

The current work handles for the first time, dispersive optical solitons in fiber Bragg gratings for the nonlinear (2+1)-dimensional Kundu–Mukherjee–Naskar equation. Two integration schemes, namely, the modified Kudryashov’s approach and the addendum to Kudryashov’s methodology are applied. Dark and bright soliton solutions as long as explicit solutions are obtained. Also, combo bright-singular solutions are introduced.


Introduction
Fiber Bragg gratings (FBGs) are of tremendous interests to physicists and engineers during the past couple of decades or more because of its applications. FBGs are applied in all optical communication systems and thus have been widely studied with several methods (Biswas et al. , 2019aBiswas 2018;Arshed et al. 2020;Ekici et al. 2019;Kudryashov et al. 2019;Kudryashov 2020aKudryashov , b, 2019aZayed et al. 2021a, b, c;Yıldırım and Mirzazadeh 2020;Trik et al. 2021;Yıldırım 2019b;Chen et al. 2020;Xia et al. 2020;He et al. 2021;Yin et al. 2020;Xu et al. 2020;. Optical nonlinearities, such as Kerr law, quadratic-cubic law, log-law, cubic-quintic-septic law, triple-power law, power-law, parabolic-law, anti-cubic law, dual-power law and parabolic-nonlocal law have been investigated by FBGs using integration technologies. Also, optical solitons in FBGs have been gained with governing models that describe the dynamics of soliton propagation, such as the Biswas (Yıldırım and Mirzazadeh 2020;Trik et al. 2021;Yıldırım 2019a, b). Actually, there are some new analytical solutions have been derived for PDEs Xia et al. 2020;. It is worth mentioning here that there are also other studies on an interesting kind of exact solutions, such as, lump solution , Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions (He et al. 2021), localized characteristics of lump and interaction solutions (Yin et al. 2020), Painleve analysis, soliton solutions, Bäcklund transformation, Laxpair and infinitely many conservation laws ) and other studies Xu et al. 2020;.
The current paper handles solitons in FBGs with the nonlinear (2+1)-dimensional Kundu-Mukherjee-Naskar equation that has solved by aid of two integration schemes. The governing model in FBGs is formulated in this the paper for the first time. Optical solitons of the model equation are revealed by the modified Kudryashov's approach (Zayed et al. 2021a, b, c) and the addendum to Kudryashov's methodology (Kudryashov 2020a;Zayed et al. 2021a, b, c). These solitons are reported in this paper for the first time. The details are sketched through, after a thorough introduction to the governing model

Governing model
The dimensionless form of the nonlinear (2+1)-dimensional Kundu-Mukherjee-Naskar model in polarization-preserving fibers is written as Ekici et al. 2019): where q(x, y, t) is a complex-valued function representing the wave profile, while a and b are real-valued constants. The first term is the linear temporal evolution, the second term represents the dispersion term, while the third term represents the nonlinearity term and i = √ −1. In Bragg gratings fibers, Eq. (1) can be written, for the first time as: and where u (x, y, t) and v(x, y, t) are complex-valued functions that represent the wave profiles, while a j , b j , c j , d j , e j , j , j , j (j = 1, 2) are real-valued constants. Here, a j are the coefficients of dispersion terms. The parameters b j , c j , d j , e j (j = 1, 2) are the coefficients of nonlinearity. Next, j , j and j give the inter-modal dispersions (IMD), the detuning parameters and the four wave mixing (4WM) parameters, respectively. Yıldırım  has discussed the birefringent fibers of Eq. (1) using the modified simple equation approach.
The main objective of this article is to apply the modified Kudryashov's approach and the addendum to Kudryashov's method to find the dark, bright and singular soliton solutions of Eqs. (2) and (3).
The organization of this article can be written as: The mathematical preliminaries are introduced in Sect. 2. In Sects. 3 and 4, we give the solutions of the system (2) and (3). In Sect. 5, the numerical simulations for some solutions are designed. In Sect. 6, conclusions are illustrated.

Mathematical preliminaries
In this section, we suppose that Eqs. (2) and (3) have the solutions: and where B 1 , B 2 , , 1 , 2 , and 0 are all non-zero real constants to be determined. The parameters B 1 and B 2 represent direct cosines and are related to the inverse widths of the soliton in the x-and y-directions, respectively, while is the soliton velocity. From the phase component, 1 and 2 give the frequencies of the solitons along the x-and y-directions, respectively, while is the wave number and 0 is the phase constant. Here, P 1 ( ) and P 2 ( ) are real valued functions which stand for the pulse shapes. If we substitute (4) and (5) into Eqs. (2) and (3) and separate the real and imaginary parts, we deduce that and Set where A is a non zero constant, such that A ≠ 1. Now, Eqs. (6)-(9) become and (4) Integrating Eqs. (13) and (14) with zero-integration constants, one gets Setting the coefficients of the linearly independent functions of Eqs. (15) and (16) to zero, yields and the constraints conditions Equations (11) and (12) are equivalent under the constraint conditions: From (21) to (22), we have the wave number of the soliton: Equation (11) can be rewritten in the form: where provided a 1 B 1 B 2 A ≠ 0. In the next two sections, we solve Eq. (25) using the following two methods: Page 5 of 13 16

The modified Kudryashov ′ s method
According to this method, we balance P �� 1 ( ) with P 3 1 ( ) in Eq. (25), we get: Now, the following cases can be considered.

Numerical simulations
In this section, we introduce the graphs of the absolute values of some solutions for Eqs. (35) and (36), Eqs. (48) and (49) and Eqs. (57) and (58). Let us now examine Figs. 1, 2 and 3. as it illustrates some of our solutions obtained in this paper. To this purpose, we choose some special values of the obtained parameters. Figure 1 shows the profile of the absolute values of the dark soliton solutions (35) and (36). Figure 2 shows the profile of the absolute values of the singular solutions (48) and (49). Figure 3 shows the profile of the absolute values of the bright soliton solutions (57) and (58).
From the above Figures, one can see that the obtained solutions possess the dark soliton, the singular and the bright soliton solutions. Also, these Figures express the behavior of these solutions which give some perspective readers how the behavior solutions are produced.