Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 19, Number 2, 2023



Article Number  40  
Number of page(s)  15  
DOI  https://doi.org/10.1051/jeos/2023038  
Published online  01 November 2023 
Research Article
Chirped gap solitons with Kudryashov’s law of selfphase modulation having dispersive reflectivity
^{1}
University of Technology and Applied Sciences, P.O. Box 14, Ibri 516, Oman
^{2}
Department of Mathematics, Sultan Qaboos University, P.O. Box 36, AlKhod, Muscat 123, Oman
^{3}
Department of Mathematics and Physics, Grambling State University, Grambling, LA71245, USA
^{4}
Mathematical Modeling and Applied Computation (MMAC) Research Group, Center of Modern Mathematical Sciences and their Applications (CMMSA), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
^{5}
Department of Applied Sciences, CrossBorder Faculty of Humanities, Economics and Engineering, Dunarea de Jos University of Galati, 111 Domneasca Street, 800201 Galati, Romania
^{6}
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa0204, Pretoria, South Africa
^{*} Corresponding author: khalil.alghafri@utas.edu.om
Received:
14
August
2023
Accepted:
12
September
2023
The present study is devoted to investigate the chirped gap solitons with Kudryashov’s law of selfphase modulation having dispersive reflectivity. Thus, the mathematical model consists of coupled nonlinear Schrödinger equation (NLSE) that describes pulse propagation in a medium of fiber Bragg gratings (BGs). To reach an integrable form for this intricate model, the phasematching condition is applied to derive equivalent equations that are handled analytically. By means of auxiliary equation method which possesses Jacobi elliptic function (JEF) solutions, various forms of soliton solutions are extracted when the modulus of JEF approaches 1. The generated chirped gap solitons have different types of structures such as bright, dark, singular, Wshaped, kink, antikink and Kinkdark solitons. Further to this, two soliton waves namely chirped bright quasisoliton and chirped dark quasisoliton are also created. The dynamic behaviors of chirped gap solitons are illustrated in addition to their corresponding chirp. It is noticed that selfphase modulation and dispersive reflectivity have remarkable influences on the pulse propagation. These detailed results may enhance the engineering applications related to the field of fiber BGs.
Key words: Chirped gap solitons / Bragg gratings / Kudryashov’s law
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The new technology in information industry depends broadly on optical fibers since its presence as a prominent mechanism that transmits light and signals over long distances and local area networks or computer networks [1–5]. The field of optical fibers can lead to further developments in the engineering and industrial applications that serve wide ranges of sectors [6, 7]. In particular, the essential applications of optical fibers include telecommunications, sensors, biomedicals, and fibre lasers [8–13]. The process of transmitting data is made by the soliton propagation due to the balance between chromatic dispersion (CD) and fiber nonlinearity. Unfortunately, the low contribution from CD may causes a restriction in the transmission scenario. This crisis can be effectively manipulated by making use of Bragg gratings (BGs) technology which compensates for low CD. In the last decade, many experts around the world have extensively studied the dynamic of solitons in fiber BGs with different forms of nonlinear refractive index such as Kerr law, quadraticcubic law, parabolic law, polynomial law, parabolicnonlocal combo law and many others [14–20]. Additionally, the characteristic of soliton propagation associated with the frequency chirp influence has been being studied continuously through the years as the chirp has significant advantages including pulse compression and amplification in optical fiber [21–27].
Rece, a significant model known as Kudryashov’s equation (KE) [28] was developed to study the soliton pulse propagation in the field of nonlinear optics. The KE is considered as a part of the family of nonlinear Schrödinger equation and it is generated from a law of refractive index. Since its appearance in 2019, the KE has been discussed by many scholars to deal with some physical features like conservation laws and optical soliton behaviors by means of various integration schemes and techniques such as Lie symmetry analysis, extended sinhGordon equation expansion method, complete discriminant system for a polynomial, new mapping method, unified auxiliary equation scheme, improved modified extended tanhfunction approach, extended trial function method and unified ansätze framework. Different soliton structures were derived such as bright, dark, singular, brightdark, singulardark solitons and others. For more details about obtained results, reader is referred to the references [29–36]. The governing KE is given by(1)where the first term represents the time evolution and . The term with the coefficient stands for the group velocity dispersion while the terms with the coefficients b_{1}, b_{2}, b_{3}, b_{4} describe the effect of selfphase modulation (SPM). In the literature, some generalized models of equation (1) are discussed to investigate optical solitons by applying distinct strategies, see as example references [37–39].
The model of KE can be also implemented to fiber BGs to examine its influence on the pulse propagation. For example, Zayed et al. [16] detected the applicability of KE to fiber BGs with dispersive reflectivity and Kerr law of nonlinear refractive index when n = 1, n = 2 n = 3. Using the extended Kudryashov’s scheme, both chirped and chirpfree optical solitons are retrieved and they are found to have the forms of dark and singular soliton profiles. It is necessary to be mentioned that the chirping associated to these solitons is expressed in terms of constant. Our current work aims to investigate the chirped gap solitons with Kudryashov’s law of selfphase modulation having dispersive reflectivity when n = 2. Herein, we assume that the chirp has a form of nonlinear function.
As stated above, this study focuses on the model of Kudryashov’s equation (KE) in fiber medium having BGs effect. The vectorcoupled KE reads [16](2) (3)where the functions q(x, t) and r(x, t) stand for forward and backward propagating waves, respectively, whereas a_{ j } for j = 1, 2 represent the coefficients of dispersive reflectivity. In the coupled equations above, b_{ j }, h_{ j }, l_{ j } and n_{ j } indicate the coefficients of selfphase modulation (SPM) and c_{ j }, d_{ j }, k_{ j }, m_{ j }, p_{ j } and s_{ j } denote the crossphase modulation XPM, respectively. The coefficients f_{ j } and g_{ j } represent the combination of SPM and XPM. Finally, α_{ j } account for intermodal dispersion and β_{ j } define detuning parameters. All of the coefficients are real valued constants and .
The following sections of paper are formatted as follows. In Section 2, the governing model is analyzed and reduced to an integrable form. Section 3 displays the derivation of chirped gap solitons with the aid of auxiliary equation method. The structures and behaviors of created solitons are discussed and described in Section 4. Finally, the summary of obtained results is given in Section 5.
2 Mathematical analysis of governing model
In order to reduce the coupledKE give by (2) and (3) to an integrable form, its complex structure is analyzed using the transformation given by(4) (5)where ξ = x − νt while ω and ν are real constants indicating the wave number and the soliton velocity. The two functions ψ_{1}(ξ) and ψ_{2}(ξ) account for the amplitudes of the solitons whereas the function ϕ(ξ) represents nonlinear phase shift. The corresponding chirp is identified as .
Inserting (4) and (5) into the coupled system (2) and (3) and breaking down into the imaginary and real components, we reach(6) (7)and(8) (9)
To handle this complexity, we assume(10)where γ ≠ 1 is a real constant. Accordingly, the set of equations (6)–(9) are converted into(11) (12)and(13) (14)
The system of equations (13) and (14) can be integrated to yield(15) (16)where ρ_{1} and ρ_{2} are the integration constants. Due to the equivalency between equations (15) and (16), one arrives at the constraint conditions given by(17) (18)
From equation (18) we come by the velocity of the soliton in the form(19)
Then, the chirp expression can be addressed as(20)
Using (15) and (16), the coupling equations (11) and (12) are changed into(21) (22)
These coupled equations are equivalent based on the conditions given by(23) (24) (25) (26) (27) (28)
Performing the balance between the terms and in equation (21) brings about the relation(29)which leads to N = 1/2. To ensure closed form solutions, we put forward the transformation of the form(30)
Upon implementing (30), equation (21) collapses into(31)where the constants σ_{ j }, (j = 0, 1, 2, 3, 4) are defined as(32)
3 Chirped gap solitons
Our target now is to derive the chirped gap solitons to the coupledKE by finding the solutions of equation (31) using a new extended auxiliary equation method [40]. This strategy provides various forms of Jacobi elliptic function solutions. To start with, we assume that equation (31) has a solution in the form(33)where η_{ j }, (j = 1, 2, 3, 4) are constants to be identified and the function F(ξ) satisfies the first order ordinary differential equation given by(34)where A_{ j }, (j = 0, 2, 4, 6) are constants to be determined. Equation (34) admits solutions having the form(35)where the function f(ξ) can be expressed in terms of the Jacobi elliptic functions (JEFs) sn(ξ, m), cn(ξ, m), dn(ξ, m) and others, where 0 < m < 1 is the modulus of JEFs that degenerate to hyperbolic functions and trigonometric functions as approaches or , respectively. Substituting (33) into equation (31) and using equation (34), we find a polynomial in terms of F′(ξ)^{ j } F(ξ)^{ l }, (j = 0, 1; l = 0, 1, …). Collecting the coefficients of terms with the same powers and equating them to zero yields a system of algebraic equations for η_{ j }, (j = 1, 2, 3, 4), A_{ j }, (j = 0, 2, 4, 6) and σ_{ j }, (j = 0, 1, 2, 3, 4). Solving this system gives us the following cases of solutions.
Family 1. If , then we arrive at the Jacobi elliptic function solutions of the coupled equations (2) and (3) as(37) (38)and(39) (40)where σ_{4} > 0, A_{6} > 0. As m → 1, solutions (37) and (38) change to the soliton solutions given by(41) (42)while solutions (39) and (40) fall into the singular soliton solutions as(43) (44)
Family 2. If , then one can obtain the Jacobi elliptic function solutions of the coupled equations (2) and (3) as(45) (46)where σ_{4} < 0, A_{6} < 0. As m → 1, solutions (45) and (46) reduce to the soliton solutions of the form(47) (48)
If , the Jacobi elliptic function solutions of the coupled equations (2) and (3) are secured as(50) (51)where A_{6} < 0. As m → 1, solutions (50) and (51) reduce to the soliton solutions of the form(52) (53)
Family 1. If , then we arrive at the Jacobi elliptic function solutions of the coupled equations (2) and (3) as(55) (56)and(57) (58)where σ_{0} <0, A_{6}> 0. When m → 1, solutions (55) and (56) become the soliton solutions given by(59) (60)while solutions (57) and (58) result in the singular soliton solutions as(61) (62)
Family 2. If , we reach the Jacobi elliptic function solutions of the coupled equations (2) and (3) as(63) (64)where σ_{0} > 0, A_{6} < 0. As m → 1, solutions (45) and (46) convert to the soliton solutions of the form(65) (66)
Family 1. If , the Jacobi elliptic function solutions of the coupled equations (2) and (3) are retrieved as(68) (69)and(70) (71)where A_{6} > 0. When m → 1, solutions (68) and (69) become the soliton solutions given by(72) (73)while solutions (70) and (71) result in the singular soliton solutions as(74) (75)
Family 2. If , we reach the Jacobi elliptic function solutions of the coupled equations (2) and (3) as(76) (77)where A_{6} < 0. When m → 1, solutions (76) and (77) turn into the soliton solutions of the form(78) (79)
Based upon the results obtained above and its counterpart in [16], the term with the parameter g_{1} has to be neglected so as to reach closed form solutions for the coupled KE, meaning that g_{1} = 0. Accordingly, equation (21) collapses to an elliptictype differential equation having the form(80)under the restriction condition(81)where σ_{2}, σ_{3} and σ_{4} are as defined in (32). Equation (80) is known to have various types of soliton solutions. One can find, for instance, a quasisoliton solution given as(82)where(83)provided that σ_{2} > 0 and to gurantee real values for the pulse width and amplitude. From this finding, the coupled equations (2) and (3) possess chirped bright quasisoliton solution in the form(84) (85)
Additionally, we can secure another form of quasisoliton solution for equation (80) as(86)where(87)under the constraint conditions(88)provided that to ensure the validity of constructing quasisoliton wave. Making use of these results, the coupled equations (2) and (3) has chirped dark quasisoliton solution presented as(89) (90)
In all solutions obtained above, the wave number ω is an arbitrary constant, the soliton velocity ν is identified in (19) and the nonlinear phase shift ϕ(ξ) can be found from (15). The chirping associated to each soliton is extracted by (20).
4 Results and discussion
As done analytically above, the implemented mathematical approach has yielded a variety of exact solutions to the coupledKE given by (2) and (3). These solutions describe distinct soliton structures for which the corresponding nonlinear chirp is expressed in terms of the reciprocal of soliton intensity. The dynamical behaviors of derived soliton waves are represented graphically to understand their physical meaning in fiber Bragg gratings medium. Thus, we illustrate the intensity profiles of gap solitons using the model parameters. The chirping associated to these solitons is also plotted.
Figure 1 displays the behaviors of solutions (41) and (42) with the model parameters given by a_{1} = a_{2} = 1, γ = α_{1} = α_{2} = ρ_{1} = m_{1} = p_{1} = 0.5, n_{1} = −0.5, s_{1} = 1.5, A_{6} = 4. Based on the change in the value of A_{4}, it can be observed that these solutions describe two soliton structures in addition to their corresponding chirp. As it can be seen from Figure 1a, the graph shows kinkdark soliton with A_{4} = 8 while Figure 1b exhibits kink wave with A_{4} = 2. We can clearly notice that Figure 2 demonstrates three forms of solitons that are deduced from solutions (47) and (48) which are plotted with same values of parameters as in Figure 1 except A_{6} = −4 and with different values of A_{4} and l_{1}. The first soliton form represents bright soliton wave as depicted in Figure 2a when A_{4} = 2; l_{1} = −0.5, −0.3, −0.1, the second soliton form describes soliton wave having W shape as shown in Figure 2b when A_{4} = 4; l_{1} = 0.3, 0.6, 1 and the third wave form is dark soliton as presented in Figure 2c when A_{4} = −2; l_{1} = −1.5, −1.2, −0.9. We have also found that solutions (52) and (53) describe three types of solitons having the former structures as shown in Figure 3 with same values of parameters as in Figure 2 and A_{4} = 8, ω = 1, β = 0.5. The bright soliton in Figure 3a is plotted with a_{1} = 1, the Wshaped soliton in 3b is plotted with a_{1} = −2.5 and the dark soliton in 3c is plotted with a_{1} = −1. In Figure 4, the graph illustrates antikink soliton characterizing solutions (59) and (60) for the values of parameters a_{1} = a_{2} = b_{1} = c_{1} = 1, γ = α_{1} = α_{2} = ρ_{1} = η_{1} = η_{3} = d_{1} = 0.5, A_{6} = 4, A_{4} = 8 while Figure 5 depicts dark soliton profile that represents solutions (65) and (66) where A_{6} = −4. Moreover, we observe that Figure 6 presents three solitonic structures describing solutions (72) and (73) for the values of parameters a_{1} = a_{2} = η_{3} = 1, γ = α_{1} = α_{2} = ρ_{1} = 0.5, A_{4} = 8, A_{6} = 4. The first structure is kinkdark soliton as displayed in Figure 6a with η_{2} = 1, η_{4} = 0.1 and η_{1} = 0.1, 0.3, 0.5. The second structure is kink soliton as plotted in Figure 6b with η_{1} = 1, η_{4} = 0.9 and η_{2} = 0.1, 0.4, 0.7. The third structure is antikink soliton as presented in Figure 2c with η_{1} = 1, η_{2} = 0.9 and η_{4} = 0.1, 0.4, 0.7. Obviously, one can see that Figure 7 demonstrates three wave forms which are bright, Wshaped and dark solitons describing solutions (78) and (79) with same values of parameters as in Figure 6 besides η_{1} = 1, A_{6} = −4. The bright soliton is shown in Figure 7a with η_{2} = 1, η_{4} = 0.1, 0.4, 0.8; the Wshaped soliton is shown in Figure 7b with η_{2} = −1.2, η_{4} = 0.1, 0.5, 1 and the dark soliton is shown in Figure 7c with η_{4} = 1, η_{2} = 0.1, 0.4, 0.8. The special case of chirped bright quasisoliton solution (84) and (85) is depicted in Figure 8 with same values of parameters as in Figure 2 and ω = 1, β = n_{1} = 0.5. Further to this, the chirped dark quasisoliton solution (89) and (90) is delineated in Figure 9 with same values of parameters as in Figure 8 and μ = 1, a_{1} = −0.5, −1.5, −2.5.
Figure 1 Soliton intensity for q(x; t) and r(x; t) given in (41) and (42) along with chirping profile. 
Figure 2 Soliton intensity for q(x; t) and r(x; t) given in (47) and (48) along with chirping profile. 
Figure 3 Soliton intensity for q(x; t) and r(x; t) given in (52) and (53) along with chirping profile. 
Figure 4 Soliton intensity for q(x; t) and r(x; t) given in (59) and (60) along with chirping profile. 
Figure 5 Soliton intensity for q(x; t) and r(x; t) given in (65) and (66) along with chirping profile. 
Figure 6 Soliton intensity for q(x; t) and r(x; t) given in (72) and (73) along with chirping profile. 
Figure 7 Soliton intensity for q(x; t) and r(x; t) given in (78) and (79) along with chirping profile. 
Figure 8 Soliton intensity for q(x; t) and r(x; t) given in (84) and (85) along with chirping profile. 
Figure 9 Soliton intensity for q(x; t) and r(x; t) given in (89) and (90) along with chirping profile. 
From the dynamical behaviors of solitons presented in Figures 1–8, it can be clearly seen that SPM causes remarkable variations in the amplitude of chirped gab solitons. On the other hand, one can notice from Figure 9 that the width of chirped dark quasisoliton is severely affected by the changes in dispersive reflectivity.
5 Conclusion
The current work concentrated on the chirped gap solitons with Kudryashov’s law of selfphase modulation having dispersive reflectivity. The medium of fiber BGs is dominated by a coupled NLSE which is reduced to an integrable form by introducing specific conditions. The extended auxiliary equation method which has solutions in terms of JEFs is applied to extract soliton solutions when the modulus of JEFs tends to 1. Due to manipulating the values of model parameters, it is found that some of solutions construct several chirped soliton structures with their corresponding chirp. The derived chirped soliton waves include bright, dark, singular, Wshaped, kink, antikink and Kinkdark solitons. In addition to this, the behaviors of solitons point out that SPM enhances the amplitude of waves. Besides, it is noticed that the width of dark quasisoliton is obviously affected by dispersive reflectivity. The results in this work could reveal important details about the dynamics of chirped gap solitons that might lead to improvements in the industrial sector related to the field of fiber BGs.
Acknowledgments
K. AlGhafri and M. Sankar gratefully acknowledge support provided by University of Technology and Applied Sciences, Ibri, Oman through the Internal Research Funding Program, grant number 01IRFPIBRI2023.
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All Figures
Figure 1 Soliton intensity for q(x; t) and r(x; t) given in (41) and (42) along with chirping profile. 

In the text 
Figure 2 Soliton intensity for q(x; t) and r(x; t) given in (47) and (48) along with chirping profile. 

In the text 
Figure 3 Soliton intensity for q(x; t) and r(x; t) given in (52) and (53) along with chirping profile. 

In the text 
Figure 4 Soliton intensity for q(x; t) and r(x; t) given in (59) and (60) along with chirping profile. 

In the text 
Figure 5 Soliton intensity for q(x; t) and r(x; t) given in (65) and (66) along with chirping profile. 

In the text 
Figure 6 Soliton intensity for q(x; t) and r(x; t) given in (72) and (73) along with chirping profile. 

In the text 
Figure 7 Soliton intensity for q(x; t) and r(x; t) given in (78) and (79) along with chirping profile. 

In the text 
Figure 8 Soliton intensity for q(x; t) and r(x; t) given in (84) and (85) along with chirping profile. 

In the text 
Figure 9 Soliton intensity for q(x; t) and r(x; t) given in (89) and (90) along with chirping profile. 

In the text 
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