Open Access
Issue
J. Eur. Opt. Society-Rapid 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

© The Author(s), published by EDP Sciences, 2023

Licence Creative CommonsThis 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 [15]. 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, bio-medicals, and fibre lasers [813]. 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, quadratic-cubic law, parabolic law, polynomial law, parabolic-nonlocal combo law and many others [1420]. 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 [2127].

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 sinh-Gordon equation expansion method, complete discriminant system for a polynomial, new mapping method, unified auxiliary equation scheme, improved modified extended tanh-function approach, extended trial function method and unified ansätze framework. Different soliton structures were derived such as bright, dark, singular, bright-dark, singular-dark solitons and others. For more details about obtained results, reader is referred to the references [2936]. The governing KE is given by i q t + a q xx + ( b 1 | q | 2 n + b 2 | q | n + b 3 | q | n + b 4 | q | 2 n ) q = 0 , $$ \mathrm{i}{q}_t+a{q}_{{xx}}+\left(\frac{{b}_1}{{\left|q\right|}^{2n}}+\frac{{b}_2}{{\left|q\right|}^n}+{b}_3{\left|q\right|}^n+{b}_4{\left|q\right|}^{2n}\right)q=0, $$(1)where the first term represents the time evolution and i = - 1 $ \mathrm{i}=\sqrt{-1}$. The term with the coefficient a $ a$ stands for the group velocity dispersion while the terms with the coefficients b1, b2, b3, b4 describe the effect of self-phase 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 [3739].

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 chirp-free 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 self-phase 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 vector-coupled KE reads [16] i q t + a 1 r xx + f 1 q b 1 | q | 4 + c 1 | q | 2 | r | 2 + d 1 | r | 4 + g 1 q h 1 | q | 2 + k 1 | r | 2 + ( l 1 | q | 2 + m 1 | r | 2 ) q + ( n 1 | q | 4 + p 1 | q | 2 | r | 2 + s 1 | r | 4 ) q + i α 1 q x + β 1 r = 0 , $$ \begin{array}{c}\mathrm{i}{q}_t+{a}_1{r}_{{xx}}+\frac{{f}_1q}{{b}_1{\left|q\right|}^4+{c}_1{\left|q\right|}^2{\left|r\right|}^2+{d}_1{\left|r\right|}^4}+\frac{{g}_1q}{{h}_1{\left|q\right|}^2+{k}_1{\left|r\right|}^2}+\left({l}_1{\left|q\right|}^2+{m}_1{\left|r\right|}^2\right)q\\ +\left({n}_1{\left|q\right|}^4+{p}_1{\left|q\right|}^2{\left|r\right|}^2+{s}_1{\left|r\right|}^4\right)q+\mathrm{i}{\alpha }_1{q}_x+{\beta }_1r=0,\end{array} $$(2) i r t + a 2 q xx + f 2 r b 2 | r | 4 + c 2 | r | 2 | q | 2 + d 2 | q | 4 + g 2 r h 2 | r | 2 + k 2 | q | 2 + ( l 2 | r | 2 + m 2 | q | 2 ) r + ( n 2 | r | 4 + p 2 | r | 2 | q | 2 + s 2 | q | 4 ) r + i α 2 r x + β 2 q = 0 , $$ \begin{array}{c}\mathrm{i}{r}_t+{a}_2{q}_{{xx}}+\frac{{f}_2r}{{b}_2{\left|r\right|}^4+{c}_2{\left|r\right|}^2{\left|q\right|}^2+{d}_2{\left|q\right|}^4}+\frac{{g}_2r}{{h}_2{\left|r\right|}^2+{k}_2{\left|q\right|}^2}+\left({l}_2{\left|r\right|}^2+{m}_2{\left|q\right|}^2\right)r\\ +\left({n}_2{\left|r\right|}^4+{p}_2{\left|r\right|}^2{\left|q\right|}^2+{s}_2{\left|q\right|}^4\right)r+\mathrm{i}{\alpha }_2{r}_x+{\beta }_2q=0,\end{array} $$(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 self-phase modulation (SPM) and c j , d j , k j , m j , p j and s j denote the cross-phase modulation XPM, respectively. The coefficients f j and g j represent the combination of SPM and XPM. Finally, α j account for inter-modal dispersion and β j define detuning parameters. All of the coefficients are real valued constants and i = - 1 $ \mathrm{i}=\sqrt{-1}$.

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 coupled-KE give by (2) and (3) to an integrable form, its complex structure is analyzed using the transformation given by q ( x , t ) = ψ 1 ( ξ ) e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\psi }_1\left(\xi \right){\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(4) r ( x , t ) = ψ 2 ( ξ ) e i ( ϕ ( ξ ) - ω t ) , $$ \begin{array}{r}\\ r\left(x,t\right)={\psi }_2\left(\xi \right){\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)},\end{array} $$(5)where ξ = x − νt while ω and ν are real constants indicating the wave number and the soliton velocity. The two functions ψ1(ξ) and ψ2(ξ) account for the amplitudes of the solitons whereas the function ϕ(ξ) represents nonlinear phase shift. The corresponding chirp is identified as δ ω ( x , t ) = - x [ ϕ ( ξ ) - ω t ] = - d ϕ ( ξ ) d ξ $ {\delta \omega }\left(x,t\right)=-\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left[\phi \left(\xi \right)-{\omega t}\right]=-\frac{\mathrm{d}\phi \left(\xi \right)}{\mathrm{d}\xi }$.

Inserting (4) and (5) into the coupled system (2) and (3) and breaking down into the imaginary and real components, we reach a 1 ψ 2 + ω ψ 1 + β 1 ψ 2 + ( ν - α 1 ) ψ 1 ϕ - a 1 ψ 2 ϕ 2 + f 1 ψ 1 b 1 ψ 1 4 + c 1 ψ 1 2 ψ 2 2 + d 1 ψ 2 4 + g 1 ψ 1 h 1 ψ 1 2 + k 1 ψ 2 2 + ( l 1 ψ 1 2 + m 1 ψ 2 2 ) ψ 1 + ( n 1 ψ 1 4 + p 1 ψ 1 2 ψ 2 2 + s 1 ψ 2 4 ) ψ 1 = 0 , $$ \begin{array}{c}{a}_1\psi {\mathrm{\prime \prime }}_2+\omega {\psi }_1+{\beta }_1{\psi }_2+\left(\nu -{\alpha }_1\right){\psi }_1\phi \mathrm{\prime}-{a}_1{\psi }_2\phi \mathrm{\prime}^2+\frac{{f}_1{\psi }_1}{{b}_1{\psi }_1^4+{c}_1{\psi }_1^2{\psi }_2^2+{d}_1{\psi }_2^4}\\ +\frac{{g}_1{\psi }_1}{{h}_1{\psi }_1^2+{k}_1{\psi }_2^2}+\left({l}_1{\psi }_1^2+{m}_1{\psi }_2^2\right){\psi }_1+\left({n}_1{\psi }_1^4+{p}_1{\psi }_1^2{\psi }_2^2+{s}_1{\psi }_2^4\right){\psi }_1=0,\end{array} $$(6) a 2 ψ 1 + ω ψ 2 + β 2 ψ 1 + ( ν - α 2 ) ψ 2 ϕ - a 2 ψ 1 ϕ 2 + f 2 ψ 2 b 2 ψ 2 4 + c 2 ψ 2 2 ψ 1 2 + d 2 ψ 1 4 + g 2 ψ 2 h 2 ψ 2 2 + k 2 ψ 1 2 + ( l 2 ψ 2 2 + m 2 ψ 1 2 ) ψ 2 + ( n 2 ψ 2 4 + p 2 ψ 2 2 ψ 1 2 + s 2 ψ 1 4 ) ψ 2 = 0 , $$ \begin{array}{c}{a}_2\psi {\mathrm{\prime \prime }}_1+\omega {\psi }_2+{\beta }_2{\psi }_1+\left(\nu -{\alpha }_2\right){\psi }_2\phi \mathrm{\prime}-{a}_2{\psi }_1\phi \mathrm{\prime}^2+\frac{{f}_2{\psi }_2}{{b}_2{\psi }_2^4+{c}_2{\psi }_2^2{\psi }_1^2+{d}_2{\psi }_1^4}\\ +\frac{{g}_2{\psi }_2}{{h}_2{\psi }_2^2+{k}_2{\psi }_1^2}+\left({l}_2{\psi }_2^2+{m}_2{\psi }_1^2\right){\psi }_2+\left({n}_2{\psi }_2^4+{p}_2{\psi }_2^2{\psi }_1^2+{s}_2{\psi }_1^4\right){\psi }_2=0,\end{array} $$(7)and ( α 1 - ν ) ψ 1 + a 1 ( ψ 2 ϕ + 2 ψ 2 ϕ ) = 0 , $$ \left({\alpha }_1-\nu \right){\psi }_1\mathrm{\prime}+{a}_1\left({\psi }_2\phi \mathrm{\prime \prime }+2{\psi }_2\mathrm{\prime}\phi \mathrm{\prime}\right)=0, $$(8) ( α 2 - ν ) ψ 2 + a 2 ( ψ 1 ϕ + 2 ψ 1 ϕ ) = 0 . $$ \left({\alpha }_2-\nu \right){\psi }_2\mathrm{\prime}+{a}_2\left({\psi }_1\phi \mathrm{\prime \prime }+2{\psi }_1\mathrm{\prime}\phi \mathrm{\prime}\right)=0. $$(9)

To handle this complexity, we assume ψ 2 = γ ψ 1 , $$ {\psi }_2=\gamma {\psi }_1, $$(10)where γ ≠ 1 is a real constant. Accordingly, the set of equations (6)(9) are converted into a 1 γ ψ 1 3 ψ 1 + f 1 b 1 + c 1 γ 2 + d 1 γ 4 + g 1 ψ 1 2 h 1 + k 1 γ 2 + [ ω + β 1 γ + ( ν - α 1 ) ϕ - a 1 γ ϕ 2 ] ψ 1 4 + ( l 1 + m 1 γ 2 ) ψ 1 6 + ( n 1 + p 1 γ 2 + s 1 γ 4 ) ψ 1 8 = 0 , $$ \begin{array}{c}{a}_1\gamma {\psi }_1^3\psi {\mathrm{\prime \prime }}_1+\frac{{f}_1}{{b}_1+{c}_1{\gamma }^2+{d}_1{\gamma }^4}+\frac{{g}_1{\psi }_1^2}{{h}_1+{k}_1{\gamma }^2}+\left[\omega +{\beta }_1\gamma +\left(\nu -{\alpha }_1\right)\phi \mathrm{\prime}-{a}_1{\gamma \phi }\mathrm{\prime}^2\right]{\psi }_1^4\\ +\left({l}_1+{m}_1{\gamma }^2\right){\psi }_1^6+\left({n}_1+{p}_1{\gamma }^2+{s}_1{\gamma }^4\right){\psi }_1^8=0,\end{array} $$(11) a 2 ψ 1 3 ψ 1 + f 2 γ b 2 γ 4 + c 2 γ 2 + d 2 + g 2 γ ψ 1 2 h 2 γ 2 + k 2 + [ ω γ + β 2 + ( ν - α 2 ) γ ϕ - a 2 ϕ 2 ] ψ 1 4 + ( l 2 γ 2 + m 2 ) γ ψ 1 6 + ( n 2 γ 4 + p 2 γ 2 + s 2 ) γ ψ 1 8 = 0 , $$ \begin{array}{c}{a}_2{\psi }_1^3\psi {\mathrm{\prime \prime }}_1+\frac{{f}_2\gamma }{{b}_2{\gamma }^4+{c}_2{\gamma }^2+{d}_2}+\frac{{g}_2\gamma {\psi }_1^2}{{h}_2{\gamma }^2+{k}_2}+\left[{\omega \gamma }+{\beta }_2+\left(\nu -{\alpha }_2\right){\gamma \phi }\mathrm{\prime}-{a}_2\phi \mathrm{\prime}^2\right]{\psi }_1^4\\ +\left({l}_2{\gamma }^2+{m}_2\right)\gamma {\psi }_1^6+\left({n}_2{\gamma }^4+{p}_2{\gamma }^2+{s}_2\right)\gamma {\psi }_1^8=0,\end{array} $$(12)and ( α 1 - ν ) ψ 1 + a 1 γ ( ψ 1 ϕ + 2 ψ 1 ϕ ) = 0 , $$ \left({\alpha }_1-\nu \right){\psi }_1\mathrm{\prime}+{a}_1\gamma \left({\psi }_1\phi \mathrm{\prime \prime }+2{\psi }_1\mathrm{\prime}\phi \mathrm{\prime}\right)=0, $$(13) ( α 2 - ν ) γ ψ 1 + a 2 ( ψ 1 ϕ + 2 ψ 1 ϕ ) = 0 . $$ \left({\alpha }_2-\nu \right)\gamma {\psi }_1\mathrm{\prime}+{a}_2\left({\psi }_1\phi \mathrm{\prime \prime }+2{\psi }_1\mathrm{\prime}\phi \mathrm{\prime}\right)=0. $$(14)

The system of equations (13) and (14) can be integrated to yield ϕ = ν - α 1 2 a 1 γ + ρ 1 ψ 1 - 2 a 1 γ , $$ {\phi }^\mathrm{\prime}=\frac{\nu -{\alpha }_1}{2{a}_1\gamma }+\frac{{\rho }_1{\psi }_1^{-2}}{{a}_1\gamma }, $$(15) ϕ = ( ν - α 2 ) γ 2 a 2 + ρ 2 ψ 1 - 2 a 2 , $$ {\phi }^\mathrm{\prime}=\frac{\left(\nu -{\alpha }_2\right)\gamma }{2{a}_2}+\frac{{\rho }_2{\psi }_1^{-2}}{{a}_2}, $$(16)where ρ1 and ρ2 are the integration constants. Due to the equivalency between equations (15) and (16), one arrives at the constraint conditions given by a 2 ρ 1 - a 1 γ ρ 2 = 0 , $$ {a}_2{\rho }_1-{a}_1\gamma {\rho }_2=0, $$(17) ( a 2 - a 1 γ 2 ) ν - ( a 2 α 1 - a 1 α 2 γ 2 ) = 0 . $$ \left({a}_2-{a}_1{\gamma }^2\right)\nu -\left({a}_2{\alpha }_1-{a}_1{\alpha }_2{\gamma }^2\right)=0. $$(18)

From equation (18) we come by the velocity of the soliton in the form ν = a 2 α 1 - a 1 α 2 γ 2 a 2 - a 1 γ 2 . $$ \nu =\frac{{a}_2{\alpha }_1-{a}_1{\alpha }_2{\gamma }^2}{{a}_2-{a}_1{\gamma }^2}. $$(19)

Then, the chirp expression can be addressed as δ ω ( x , t ) = - [ ν - α 1 2 a 1 γ + ρ 1 ψ 1 - 2 γ a 1 ] . $$ {\delta \omega }\left(x,t\right)=-\left[\frac{\nu -{\alpha }_1}{2{a}_1\gamma }+\frac{{\rho }_1{\psi }_1^{-2}}{\gamma {a}_1}\right]. $$(20)

Using (15) and (16), the coupling equations (11) and (12) are changed into a 1 γ ψ 1 3 ψ 1 - ρ 1 2 a 1 γ + f 1 b 1 + c 1 γ 2 + d 1 γ 4 + g 1 ψ 1 2 h 1 + k 1 γ 2 + [ ω + β 1 γ + ( ν - α 1 ) 2 4 a 1 γ ] ψ 1 4 + ( l 1 + m 1 γ 2 ) ψ 1 6 + ( n 1 + p 1 γ 2 + s 1 γ 4 ) ψ 1 8 = 0 , $$ \begin{array}{c}{a}_1\gamma {\psi }_1^3{\psi }_1^{\mathrm{\prime \prime }}-\frac{{\rho }_1^2}{{a}_1\gamma }+\frac{{f}_1}{{b}_1+{c}_1{\gamma }^2+{d}_1{\gamma }^4}+\frac{{g}_1{\psi }_1^2}{{h}_1+{k}_1{\gamma }^2}+\left[\omega +{\beta }_1\gamma +\frac{{\left(\nu -{\alpha }_1\right)}^2}{4{a}_1\gamma }\right]{\psi }_1^4\\ +\left({l}_1+{m}_1{\gamma }^2\right){\psi }_1^6+\left({n}_1+{p}_1{\gamma }^2+{s}_1{\gamma }^4\right){\psi }_1^8=0,\end{array} $$(21) a 2 ψ 1 3 ψ 1 - a 2 ρ 1 2 a 1 2 γ 2 + f 2 γ b 2 γ 4 + c 2 γ 2 + d 2 + g 2 γ ψ 1 2 h 2 γ 2 + k 2 + [ ω γ + β 2 + ( ν - α 2 ) 2 γ 2 4 a 2 ] ψ 1 4 + ( l 2 γ 2 + m 2 ) γ ψ 1 6 + ( n 2 γ 4 + p 2 γ 2 + s 2 ) γ ψ 1 8 = 0 . $$ \begin{array}{c}{a}_2{\psi }_1^3{\psi }_1^{\mathrm{\prime \prime }}-\frac{{a}_2{\rho }_1^2}{{a}_1^2{\gamma }^2}+\frac{{f}_2\gamma }{{b}_2{\gamma }^4+{c}_2{\gamma }^2+{d}_2}+\frac{{g}_2\gamma {\psi }_1^2}{{h}_2{\gamma }^2+{k}_2}+\left[{\omega \gamma }+{\beta }_2+\frac{{\left(\nu -{\alpha }_2\right)}^2{\gamma }^2}{4{a}_2}\right]{\psi }_1^4\\ +\left({l}_2{\gamma }^2+{m}_2\right)\gamma {\psi }_1^6+\left({n}_2{\gamma }^4+{p}_2{\gamma }^2+{s}_2\right)\gamma {\psi }_1^8=0.\end{array} $$(22)

These coupled equations are equivalent based on the conditions given by a 2 = a 1 γ , $$ {a}_2={a}_1\gamma, $$(23) f 2 γ ( b 1 + c 1 γ 2 + d 1 γ 4 ) = f 1 ( b 2 γ 4 + c 2 γ 2 + d 2 ) , $$ {f}_2\gamma \left({b}_1+{c}_1{\gamma }^2+{d}_1{\gamma }^4\right)={f}_1\left({b}_2{\gamma }^4+{c}_2{\gamma }^2+{d}_2\right), $$(24) g 2 γ ( h 1 + k 1 γ 2 ) = g 1 ( h 2 γ 2 + k 2 ) , $$ {g}_2\gamma \left({h}_1+{k}_1{\gamma }^2\right)={g}_1\left({h}_2{\gamma }^2+{k}_2\right), $$(25) 4 a 1 γ ( ω γ + β 2 ) + ( ν - α 2 ) 2 γ 2 = 4 a 1 γ ( ω + β 1 γ ) + ( ν - α 1 ) 2 , $$ 4{a}_1\gamma \left({\omega \gamma }+{\beta }_2\right)+{\left(\nu -{\alpha }_2\right)}^2{\gamma }^2=4{a}_1\gamma \left(\omega +{\beta }_1\gamma \right)+{\left(\nu -{\alpha }_1\right)}^2, $$(26) ( l 2 γ 2 + m 2 ) γ = l 1 + m 1 γ 2 , $$ \left({l}_2{\gamma }^2+{m}_2\right)\gamma ={l}_1+{m}_1{\gamma }^2, $$(27) ( n 2 γ 4 + p 2 γ 2 + s 2 ) γ = n 1 + p 1 γ 2 + s 1 γ 4 . $$ \left({n}_2{\gamma }^4+{p}_2{\gamma }^2+{s}_2\right)\gamma ={n}_1+{p}_1{\gamma }^2+{s}_1{\gamma }^4. $$(28)

Performing the balance between the terms ψ 1 3 ψ 1 $ {\psi }_1^3\psi {\mathrm{\prime \prime }}_1$ and ψ 1 8 $ {\psi }_1^8$ in equation (21) brings about the relation 4 N + 2 = 8 N , $$ 4N+2=8N, $$(29)which leads to N = 1/2. To ensure closed form solutions, we put forward the transformation of the form ψ 1 ( ξ ) = P 1 2 ( ξ ) . $$ {\psi }_1\left(\xi \right)={P}^{\frac{1}{2}}\left(\xi \right). $$(30)

Upon implementing (30), equation (21) collapses into P 2 - 2 PP + σ 0 + σ 1 P + σ 2 P 2 + σ 3 P 3 + σ 4 P 4 = 0 , $$ P\mathrm{\prime}^2-2{PP}\mathrm{\prime \prime }+{\sigma }_0+{\sigma }_1P+{\sigma }_2{P}^2+{\sigma }_3{P}^3+{\sigma }_4{P}^4=0, $$(31)where the constants σ j , (j = 0, 1, 2, 3, 4) are defined as σ 0 = 4 ρ 1 2 a 1 2 γ 2 - 4 f 1 a 1 γ ( b 1 + c 1 γ 2 + d 1 γ 4 ) ,   σ 1 = - 4 g 1 a 1 γ ( h 1 + k 1 γ 2 ) , σ 2 = - 4 a 1 γ ( ω + β 1 γ ) + ( ν - α 1 ) 2 a 1 2 γ 2 , σ 3 = - 4 ( l 1 + m 1 γ 2 ) a 1 γ , σ 4 = - 4 ( n 1 + p 1 γ 2 + s 1 γ 4 ) a 1 γ . $$ \begin{array}{rr}& {\sigma }_0=\frac{4{\rho }_1^2}{{a}_1^2{\gamma }^2}-\frac{4{f}_1}{{a}_1\gamma \left({b}_1+{c}_1{\gamma }^2+{d}_1{\gamma }^4\right)},\enspace \hspace{1em}{\sigma }_1=\frac{-4{g}_1}{{a}_1\gamma \left({h}_1+{k}_1{\gamma }^2\right)},\\ & {\sigma }_2=-\frac{4{a}_1\gamma \left(\omega +{\beta }_1\gamma \right)+{\left(\nu -{\alpha }_1\right)}^2}{{a}_1^2{\gamma }^2},\hspace{1em}{\sigma }_3=-\frac{4\left({l}_1+{m}_1{\gamma }^2\right)}{{a}_1\gamma },\\ & {\sigma }_4=-\frac{4\left({n}_1+{p}_1{\gamma }^2+{s}_1{\gamma }^4\right)}{{a}_1\gamma }.\end{array} $$(32)

3 Chirped gap solitons

Our target now is to derive the chirped gap solitons to the coupled-KE 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 P ( ξ ) = η 1 + η 2 F 2 ( ξ ) η 3 + η 4 F 2 ( ξ ) , $$ P\left(\xi \right)=\frac{{\eta }_1+{\eta }_2{F}^2\left(\xi \right)}{{\eta }_3+{\eta }_4{F}^2\left(\xi \right)}, $$(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 ( F ( ξ ) ) 2 = A 0 + A 2 F ( ξ ) 2 + A 4 F ( ξ ) 4 + A 6 F ( ξ ) 6 , $$ {\left(F\mathrm{\prime}\left(\xi \right)\right)}^2={A}_0+{A}_2F{\left(\xi \right)}^2+{A}_4F{\left(\xi \right)}^4+{A}_6F{\left(\xi \right)}^6, $$(34)where A j , (j = 0, 2, 4, 6) are constants to be determined. Equation (34) admits solutions having the form F ( ξ ) = 1 2 [ - A 4 A 6 ( 1 ± f ( ξ ) ) ] 1 2 , $$ F\left(\xi \right)=\frac{1}{2}{\left[-\frac{{A}_4}{{A}_6}\left(1\pm f\left(\xi \right)\right)\right]}^{\frac{1}{2}}, $$(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 m $ m$ approaches 1 $ 1$ or 0 $ 0$, 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.

Case 1. η 4 = 0 , η 1 = η 2 ( 8 A 4 η 3 - σ 3 η 2 ) 32 A 6 η 3 , η 2 = ± η 3 12 A 6 σ 4 , σ 0 = η 1 ( 8 A 0 η 2 3 η 3 - 8 A 2 η 1 η 2 2 η 3 + 24 A 6 η 1 3 η 3 + σ 3 η 1 2 η 2 2 ) 2 η 2 2 η 3 3 , σ 1 = 0 , σ 2 = 8 A 2 η 2 2 η 3 - 48 A 6 η 1 2 η 3 - 3 σ 3 η 2 2 η 1 2 η 2 2 η 3 . $$ \begin{array}{rr}& {\eta }_4=0,\hspace{1em}{\eta }_1=\frac{{\eta }_2\left(8{A}_4{\eta }_3-{\sigma }_3{\eta }_2\right)}{32{A}_6{\eta }_3},\hspace{1em}{\eta }_2=\pm {\eta }_3\sqrt{\frac{12{A}_6}{{\sigma }_4}},\\ & {\sigma }_0=\frac{{\eta }_1\left(8{A}_0{\eta }_2^3{\eta }_3-8{A}_2{\eta }_1{\eta }_2^2{\eta }_3+24{A}_6{\eta }_1^3{\eta }_3+{\sigma }_3{\eta }_1^2{\eta }_2^2\right)}{2{\eta }_2^2{\eta }_3^3},\\ & {\sigma }_1=0,\hspace{1em}{\sigma }_2=\frac{8{A}_2{\eta }_2^2{\eta }_3-48{A}_6{\eta }_1^2{\eta }_3-3{\sigma }_3{\eta }_2^2{\eta }_1}{2{\eta }_2^2{\eta }_3}.\end{array} $$(36)

Family 1. If A 0 = A 4 3 ( m 2 - 1 ) 32 A 6 2 m 2 , A 2 = A 4 2 ( 5 m 2 - 1 ) 16 A 6 m 2 $ {A}_0=\frac{{A}_4^3\left({m}^2-1\right)}{32{A}_6^2{m}^2},{A}_2=\frac{{A}_4^2\left(5{m}^2-1\right)}{16{A}_6{m}^2}$, then we arrive at the Jacobi elliptic function solutions of the coupled equations (2) and (3) as q ( x , t ) = { 1 8 A 6 σ 4 [ - 3 A 6 σ 3 ± 4 A 4 3 A 6 σ 4 sn ( A 4 2 m 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{1}{8{A}_6{\sigma }_4}\left[-3{A}_6{\sigma }_3\pm 4{A}_4\sqrt{3{A}_6{\sigma }_4}\mathrm{sn}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(37) r ( x , t ) = γ q ( x , t ) , $$ \begin{array}{rrr}& & \\ & & r\left(x,t\right)={\gamma q}\left(x,t\right),\end{array} $$(38)and q ( x , t ) = { 1 8 m A 6 σ 4 [ - 3 m A 6 σ 3 ± 4 A 4 3 A 6 σ 4 ns ( A 4 2 m 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{1}{8m{A}_6{\sigma }_4}\left[-3m{A}_6{\sigma }_3\pm 4{A}_4\sqrt{3{A}_6{\sigma }_4}\mathrm{ns}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(39) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(40)where σ4 > 0, A6 > 0. As m → 1, solutions (37) and (38) change to the soliton solutions given by q ( x , t ) = { 1 8 A 6 σ 4 [ - 3 A 6 σ 3 ± 4 A 4 3 A 6 σ 4 tanh ( A 4 2 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{1}{8{A}_6{\sigma }_4}\left[-3{A}_6{\sigma }_3\pm 4{A}_4\sqrt{3{A}_6{\sigma }_4}\mathrm{tanh}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(41) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(42)while solutions (39) and (40) fall into the singular soliton solutions as q ( x , t ) = { 1 8 A 6 σ 4 [ - 3 A 6 σ 3 ± 4 A 4 3 A 6 σ 4 coth ( A 4 2 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{1}{8{A}_6{\sigma }_4}\left[-3{A}_6{\sigma }_3\pm 4{A}_4\sqrt{3{A}_6{\sigma }_4}\mathrm{coth}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(43) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(44)

Family 2. If A 0 = A 4 3 32 A 6 2 m 2 , A 2 = A 4 2 ( 4 m 2 + 1 ) 16 A 6 m 2 $ {A}_0=\frac{{A}_4^3}{32{A}_6^2{m}^2},{A}_2=\frac{{A}_4^2\left(4{m}^2+1\right)}{16{A}_6{m}^2}$, then one can obtain the Jacobi elliptic function solutions of the coupled equations (2) and (3) as q ( x , t ) = { 1 8 A 6 σ 4 [ - 3 A 6 σ 3 ± 4 A 4 3 A 6 σ 4 cn ( A 4 2 m - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{1}{8{A}_6{\sigma }_4}\left[-3{A}_6{\sigma }_3\pm 4{A}_4\sqrt{3{A}_6{\sigma }_4}\mathrm{cn}\left(\frac{{A}_4}{2m}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(45) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(46)where σ4 < 0, A6 < 0. As m → 1, solutions (45) and (46) reduce to the soliton solutions of the form q ( x , t ) = { 1 8 A 6 σ 4 [ - 3 A 6 σ 3 ± 4 A 4 3 A 6 σ 4 sech ( A 4 2 - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{1}{8{A}_6{\sigma }_4}\left[-3{A}_6{\sigma }_3\pm 4{A}_4\sqrt{3{A}_6{\sigma }_4}\mathrm{sech}\left(\frac{{A}_4}{2}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(47) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(48)

Case 2. η 3 = 0 , η 2 = ( 4 A 2 - σ 2 ) η 1 12 A 0 , η 4 = σ 3 η 1 8 A 0 , σ 0 = 4 ( A 0 η 2 3 - A 2 η 1 η 2 2 + A 4 η 1 2 η 2 - A 6 η 1 3 ) η 1 η 4 2 , σ 1 = 0 , σ 4 = 0 . $$ \begin{array}{rr}& {\eta }_3=0,\hspace{1em}{\eta }_2=\frac{\left(4{A}_2-{\sigma }_2\right)\eta 1}{12{A}_0},\hspace{1em}{\eta }_4=\frac{{\sigma }_3{\eta }_1}{8{A}_0},\\ & {\sigma }_0=\frac{4\left({A}_0{\eta }_2^3-{A}_2{\eta }_1{\eta }_2^2+{A}_4{\eta }_1^2{\eta }_2-{A}_6{\eta }_1^3\right)}{{\eta }_1{\eta }_4^2},\hspace{1em}{\sigma }_1=0,\hspace{1em}{\sigma }_4=0.\end{array} $$(49)

If A 0 = A 4 3 32 A 6 2 m 2 , A 2 = A 4 2 ( 4 m 2 + 1 ) 16 A 6 m 2 $ {A}_0=\frac{{A}_4^3}{32{A}_6^2{m}^2},{A}_2=\frac{{A}_4^2\left(4{m}^2+1\right)}{16{A}_6{m}^2}$, the Jacobi elliptic function solutions of the coupled equations (2) and (3) are secured as q ( x , t ) = { ( A 4 2 + 4 m 2 [ A 4 2 - A 6 σ 2 ] ) [ 1 + cn ( A 4 2 m - 1 A 6 ( x - ν t ) ) ] - 6 A 4 2 6 m 2 A 6 σ 3 [ 1 + cn ( A 4 2 m - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{\left({A}_4^2+4{m}^2\left[{A}_4^2-{A}_6{\sigma }_2\right]\right)\left[1+\mathrm{cn}\left(\frac{{A}_4}{2m}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]-6{A}_4^2}{6{m}^2{A}_6{\sigma }_3\left[1+\mathrm{cn}\left(\frac{{A}_4}{2m}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(50) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(51)where A6 < 0. As m → 1, solutions (50) and (51) reduce to the soliton solutions of the form q ( x , t ) = { ( 5 A 4 2 - 4 A 6 σ 2 ) [ 1 + sech ( A 4 2 - 1 A 6 ( x - ν t ) ) ] - 6 A 4 2 6 A 6 σ 3 [ 1 + sech ( A 4 2 - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{\left(5{A}_4^2-4{A}_6{\sigma }_2\right)\left[1+\mathrm{sech}\left(\frac{{A}_4}{2}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]-6{A}_4^2}{6{A}_6{\sigma }_3\left[1+\mathrm{sech}\left(\frac{{A}_4}{2}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(52) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(53)

Case 3. η 2 = 0 , η 4 = ± 2 η 1 σ 0 - A 6 σ 0 , σ 1 = 0 , σ 2 = 12 A 0 η 3 η 4 3 - 8 A 2 η 3 2 η 4 2 + 12 A 6 η 3 4 + σ 4 η 1 2 η 4 2 η 3 2 η 4 2 , σ 3 = - ( 16 A 0 η 3 η 4 3 - 8 A 2 η 3 2 η 4 2 + 8 A 6 η 3 4 + 2 σ 4 η 1 2 η 4 2 ) η 1 η 4 2 η 3 , σ 4 = 12 η 3 ( - A 0 η 4 3 + A 2 η 3 η 4 2 - A 4 η 3 2 η 4 + A 6 η 3 3 ) η 4 2 η 1 2 . $$ \begin{array}{rr}& {\eta }_2=0,\hspace{1em}{\eta }_4=\pm \frac{2{\eta }_1}{{\sigma }_0}\sqrt{-{A}_6{\sigma }_0},\hspace{1em}{\sigma }_1=0,\\ & {\sigma }_2=\frac{12{A}_0{\eta }_3{\eta }_4^3-8{A}_2{\eta }_3^2{\eta }_4^2+12{A}_6{\eta }_3^4+{\sigma }_4{\eta }_1^2{\eta }_4^2}{{\eta }_3^2{\eta }_4^2},\\ & {\sigma }_3=\frac{-\left(16{A}_0{\eta }_3{\eta }_4^3-8{A}_2{\eta }_3^2{\eta }_4^2+8{A}_6{\eta }_3^4+2{\sigma }_4{\eta }_1^2{\eta }_4^2\right)}{{\eta }_1{\eta }_4^2{\eta }_3},\\ & {\sigma }_4=\frac{12{\eta }_3\left(-{A}_0{\eta }_4^3+{A}_2{\eta }_3{\eta }_4^2-{A}_4{\eta }_3^2{\eta }_4+{A}_6{\eta }_3^3\right)}{{\eta }_4^2{\eta }_1^2}.\end{array} $$(54)

Family 1. If A 0 = A 4 3 ( m 2 - 1 ) 32 A 6 2 m 2 , A 2 = A 4 2 ( 5 m 2 - 1 ) 16 A 6 m 2 $ {A}_0=\frac{{A}_4^3\left({m}^2-1\right)}{32{A}_6^2{m}^2},{A}_2=\frac{{A}_4^2\left(5{m}^2-1\right)}{16{A}_6{m}^2}$, then we arrive at the Jacobi elliptic function solutions of the coupled equations (2) and (3) as q ( x , t ) = { 2 A 6 σ 0 η 1 2 A 6 σ 0 η 3 A 4 η 1 - A 6 σ 0 [ 1 + sn ( A 4 2 m 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{2{A}_6{\sigma }_0{\eta }_1}{2{A}_6{\sigma }_0{\eta }_3\mp {A}_4{\eta }_1\sqrt{-{A}_6{\sigma }_0}\left[1+\mathrm{sn}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(55) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(56)and q ( x , t ) = { 2 m A 6 σ 0 η 1 2 m A 6 σ 0 η 3 A 4 η 1 - A 6 σ 0 [ m + ns ( A 4 2 m 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{2m{A}_6{\sigma }_0{\eta }_1}{2m{A}_6{\sigma }_0{\eta }_3\mp {A}_4{\eta }_1\sqrt{-{A}_6{\sigma }_0}\left[m+\mathrm{ns}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(57) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(58)where σ0 <0, A6> 0. When m → 1, solutions (55) and (56) become the soliton solutions given by q ( x , t ) = { 2 A 6 σ 0 η 1 2 A 6 σ 0 η 3 A 4 η 1 - A 6 σ 0 [ 1 + tanh ( A 4 2 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{2{A}_6{\sigma }_0{\eta }_1}{2{A}_6{\sigma }_0{\eta }_3\mp {A}_4{\eta }_1\sqrt{-{A}_6{\sigma }_0}\left[1+\mathrm{tanh}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(59) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(60)while solutions (57) and (58) result in the singular soliton solutions as q ( x , t ) = { 2 A 6 σ 0 η 1 2 A 6 σ 0 η 3 A 4 η 1 - A 6 σ 0 [ 1 + coth ( A 4 2 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{2{A}_6{\sigma }_0{\eta }_1}{2{A}_6{\sigma }_0{\eta }_3\mp {A}_4{\eta }_1\sqrt{-{A}_6{\sigma }_0}\left[1+\mathrm{coth}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(61) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(62)

Family 2. If A 0 = A 4 3 32 A 6 2 m 2 , A 2 = A 4 2 ( 4 m 2 + 1 ) 16 A 6 m 2 $ {A}_0=\frac{{A}_4^3}{32{A}_6^2{m}^2},{A}_2=\frac{{A}_4^2\left(4{m}^2+1\right)}{16{A}_6{m}^2}$, we reach the Jacobi elliptic function solutions of the coupled equations (2) and (3) as q ( x , t ) = { 2 A 6 σ 0 η 1 2 A 6 σ 0 η 3 A 4 η 1 - A 6 σ 0 [ 1 + cn ( A 4 2 m - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{2{A}_6{\sigma }_0{\eta }_1}{2{A}_6{\sigma }_0{\eta }_3\mp {A}_4{\eta }_1\sqrt{-{A}_6{\sigma }_0}\left[1+\mathrm{cn}\left(\frac{{A}_4}{2m}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(63) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(64)where σ0 > 0, A6 < 0. As m → 1, solutions (45) and (46) convert to the soliton solutions of the form q ( x , t ) = { 2 A 6 σ 0 η 1 2 A 6 σ 0 η 3 A 4 η 1 - A 6 σ 0 [ 1 + sech ( A 4 2 - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{2{A}_6{\sigma }_0{\eta }_1}{2{A}_6{\sigma }_0{\eta }_3\mp {A}_4{\eta }_1\sqrt{-{A}_6{\sigma }_0}\left[1+\mathrm{sech}\left(\frac{{A}_4}{2}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(65) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(66)

Case 4. σ 1 = 0 , σ 0 = 4 η 1 ( A 0 η 2 3 - A 2 η 1 η 2 2 + A 4 η 1 2 η 2 - A 6 η 1 3 ) ( η 1 η 4 - η 2 η 3 ) 2 , σ 4 = - 12 η 3 ( A 0 η 4 3 - A 2 η 3 η 4 2 + A 4 η 3 2 η 4 - A 6 η 3 3 ) ( η 1 η 4 - η 2 η 3 ) 2 , σ 2 = - ( 24 A 0 η 3 η 4 - 8 A 2 η 3 2 + 4 η 1 2 σ 4 + 3 η 1 η 3 σ 3 ) 2 η 3 2 , σ 3 = - 8 ( 9 A 0 η 3 η 4 2 - 6 A 2 η 3 2 η 4 + 3 A 4 η 3 3 + η 1 2 η 4 σ 4 - η 1 η 2 η 3 σ 4 ) 3 η 3 ( η 1 η 4 - η 2 η 3 ) . $$ \begin{array}{rr}& {\sigma }_1=0,\hspace{1em}{\sigma }_0=\frac{4{\eta }_1\left({A}_0{\eta }_2^3-{A}_2{\eta }_1{\eta }_2^2+{A}_4{\eta }_1^2{\eta }_2-{A}_6{\eta }_1^3\right)}{{\left({\eta }_1{\eta }_4-{\eta }_2{\eta }_3\right)}^2},\\ & {\sigma }_4=\frac{-12{\eta }_3\left({A}_0{\eta }_4^3-{A}_2{\eta }_3{\eta }_4^2+{A}_4{\eta }_3^2{\eta }_4-{A}_6{\eta }_3^3\right)}{{\left({\eta }_1{\eta }_4-{\eta }_2{\eta }_3\right)}^2},\\ & {\sigma }_2=\frac{-\left(24{A}_0{\eta }_3{\eta }_4-8{A}_2{\eta }_3^2+4{\eta }_1^2{\sigma }_4+3{\eta }_1{\eta }_3{\sigma }_3\right)}{2{\eta }_3^2},\\ & {\sigma }_3=\frac{-8\left(9{A}_0{\eta }_3{\eta }_4^2-6{A}_2{\eta }_3^2{\eta }_4+3{A}_4{\eta }_3^3+{\eta }_1^2{\eta }_4{\sigma }_4-{\eta }_1{\eta }_2{\eta }_3{\sigma }_4\right)}{3{\eta }_3\left({\eta }_1{\eta }_4-{\eta }_2{\eta }_3\right)}.\end{array} $$(67)

Family 1. If A 0 = A 4 3 ( m 2 - 1 ) 32 A 6 2 m 2 , A 2 = A 4 2 ( 5 m 2 - 1 ) 16 A 6 m 2 $ {A}_0=\frac{{A}_4^3\left({m}^2-1\right)}{32{A}_6^2{m}^2},{A}_2=\frac{{A}_4^2\left(5{m}^2-1\right)}{16{A}_6{m}^2}$, the Jacobi elliptic function solutions of the coupled equations (2) and (3) are retrieved as q ( x , t ) = { 4 A 6 η 1 - A 4 η 2 [ 1 + sn ( A 4 2 m 1 A 6 ( x - ν t ) ) ] 4 A 6 η 3 - A 4 η 4 [ 1 + sn ( A 4 2 m 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{4{A}_6{\eta }_1-{A}_4{\eta }_2\left[1+\mathrm{sn}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}{4{A}_6{\eta }_3-{A}_4{\eta }_4\left[1+\mathrm{sn}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(68) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(69)and q ( x , t ) = { 4 m A 6 η 1 - A 4 η 2 [ m + ns ( A 4 2 m 1 A 6 ( x - ν t ) ) ] 4 m A 6 η 3 - A 4 η 4 [ m + ns ( A 4 2 m 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{4m{A}_6{\eta }_1-{A}_4{\eta }_2\left[m+\mathrm{ns}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}{4m{A}_6{\eta }_3-{A}_4{\eta }_4\left[m+\mathrm{ns}\left(\frac{{A}_4}{2m}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(70) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(71)where A6 > 0. When m → 1, solutions (68) and (69) become the soliton solutions given by q ( x , t ) = { 4 A 6 η 1 - A 4 η 2 [ 1 + tanh ( A 4 2 1 A 6 ( x - ν t ) ) ] 4 A 6 η 3 - A 4 η 4 [ 1 + tanh ( A 4 2 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{4{A}_6{\eta }_1-{A}_4{\eta }_2\left[1+\mathrm{tanh}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}{4{A}_6{\eta }_3-{A}_4{\eta }_4\left[1+\mathrm{tanh}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(72) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(73)while solutions (70) and (71) result in the singular soliton solutions as q ( x , t ) = { 4 A 6 η 1 - A 4 η 2 [ 1 + coth ( A 4 2 1 A 6 ( x - ν t ) ) ] 4 A 6 η 3 - A 4 η 4 [ 1 + coth ( A 4 2 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{4{A}_6{\eta }_1-{A}_4{\eta }_2\left[1+\mathrm{coth}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}{4{A}_6{\eta }_3-{A}_4{\eta }_4\left[1+\mathrm{coth}\left(\frac{{A}_4}{2}\sqrt{\frac{1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(74) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(75)

Family 2. If A 0 = A 4 3 32 A 6 2 m 2 , A 2 = A 4 2 ( 4 m 2 + 1 ) 16 A 6 m 2 $ {A}_0=\frac{{A}_4^3}{32{A}_6^2{m}^2},{A}_2=\frac{{A}_4^2\left(4{m}^2+1\right)}{16{A}_6{m}^2}$, we reach the Jacobi elliptic function solutions of the coupled equations (2) and (3) as q ( x , t ) = { 4 A 6 η 1 - A 4 η 2 [ 1 + cn ( A 4 2 m - 1 A 6 ( x - ν t ) ) ] 4 A 6 η 3 - A 4 η 4 [ 1 + cn ( A 4 2 m - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{4{A}_6{\eta }_1-{A}_4{\eta }_2\left[1+\mathrm{cn}\left(\frac{{A}_4}{2m}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}{4{A}_6{\eta }_3-{A}_4{\eta }_4\left[1+\mathrm{cn}\left(\frac{{A}_4}{2m}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(76) r ( x , t ) = γ q ( x , t ) , $$ r\left(x,t\right)={\gamma q}\left(x,t\right), $$(77)where A6 < 0. When m → 1, solutions (76) and (77) turn into the soliton solutions of the form q ( x , t ) = { 4 A 6 η 1 - A 4 η 2 [ 1 + sech ( A 4 2 - 1 A 6 ( x - ν t ) ) ] 4 A 6 η 3 - A 4 η 4 [ 1 + sech ( A 4 2 - 1 A 6 ( x - ν t ) ) ] } 1 2 e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)={\left\{\frac{4{A}_6{\eta }_1-{A}_4{\eta }_2\left[1+\mathrm{sech}\left(\frac{{A}_4}{2}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}{4{A}_6{\eta }_3-{A}_4{\eta }_4\left[1+\mathrm{sech}\left(\frac{{A}_4}{2}\sqrt{\frac{-1}{{A}_6}}\left(x-{\nu t}\right)\right)\right]}\right\}}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(78) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(79)

Based upon the results obtained above and its counterpart in [16], the term with the parameter g1 has to be neglected so as to reach closed form solutions for the coupled KE, meaning that g1 = 0. Accordingly, equation (21) collapses to an elliptic-type differential equation having the form ψ 1 - σ 2 4 ψ 1 - σ 3 4 ψ 1 3 - σ 4 4 ψ 1 5 = 0 , $$ {\psi }_1^{\mathrm{\prime \prime }}-\frac{{\sigma }_2}{4}{\psi }_1-\frac{{\sigma }_3}{4}{\psi }_1^3-\frac{{\sigma }_4}{4}{\psi }_1^5=0, $$(80)under the restriction condition f 1 a 1 γ = ρ 1 2 ( b 1 + c 1 γ 2 + d 1 γ 4 ) , $$ {f}_1{a}_1\gamma ={\rho }_1^2\left({b}_1+{c}_1{\gamma }^2+{d}_1{\gamma }^4\right), $$(81)where σ2, σ3 and σ4 are as defined in (32). Equation (80) is known to have various types of soliton solutions. One can find, for instance, a quasi-soliton solution given as ψ 1 ( ξ ) = κ 1 sech ( Ω ξ ) 1 + κ 2 sech 2 ( Ω ξ ) , $$ {\psi }_1\left(\xi \right)=\frac{{\kappa }_1\mathrm{sech}\left(\mathrm{\Omega }\xi \right)}{\sqrt{1+{\kappa }_2{\mathrm{sech}}^2\left(\mathrm{\Omega }\xi \right)}}, $$(82)where Ω = 1 2 σ 2 ,   κ 1 4 = ( 12 σ 2 2 3 σ 3 2 - 16 σ 2 σ 4 ) , κ 2 = - 1 2 ( 1 + σ 3 3 3 σ 3 2 - 16 σ 2 σ 4 ) , $$ \mathrm{\Omega }=\frac{1}{2}\sqrt{{\sigma }_2},\hspace{1em}\enspace {\kappa }_1^4=\left(\frac{12{\sigma }_2^2}{3{\sigma }_3^2-16{\sigma }_2{\sigma }_4}\right),\hspace{1em}{\kappa }_2=-\frac{1}{2}\left(1+{\sigma }_3\sqrt{\frac{3}{3{\sigma }_3^2-16{\sigma }_2{\sigma }_4}}\right), $$(83)provided that σ2 > 0 and 3 σ 3 2 - 16 σ 2 σ 4 > 0 $ 3{\sigma }_3^2-16{\sigma }_2{\sigma }_4>0$ to gurantee real values for the pulse width and amplitude. From this finding, the coupled equations (2) and (3) possess chirped bright quasi-soliton solution in the form q ( x , t ) = κ 1 sech [ Ω ( x - ν t ) ] 1 + κ 2 sech 2 [ Ω ( x - ν t ) ] e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)=\frac{{\kappa }_1\mathrm{sech}\left[\mathrm{\Omega }\left(x-{\nu t}\right)\right]}{\sqrt{1+{\kappa }_2{\mathrm{sech}}^2\left[\mathrm{\Omega }\left(x-{\nu t}\right)\right]}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(84) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(85)

Additionally, we can secure another form of quasi-soliton solution for equation (80) as ψ 1 ( ξ ) = μ 1 tanh ( Λ ξ ) 1 + μ 2 sech 2 ( Λ ξ ) , $$ {\psi }_1\left(\xi \right)=\frac{{\mu }_1\mathrm{tanh}\left({\Lambda \xi }\right)}{\sqrt{1+{\mu }_2{\mathrm{sech}}^2\left({\Lambda \xi }\right)}}, $$(86)where Λ = μ 1 4 2 σ 3 + 4 σ 4 μ 1 2 , μ 2 = 2 σ 4 μ 1 2 3 σ 3 + 4 σ 4 μ 1 2 , $$ \mathrm{\Lambda }=\frac{{\mu }_1}{4}\sqrt{2{\sigma }_3+4{\sigma }_4{\mu }_1^2},\hspace{1em}{\mu }_2=\frac{2{\sigma }_4{\mu }_1^2}{3{\sigma }_3+4{\sigma }_4{\mu }_1^2}, $$(87)under the constraint conditions σ 2 + σ 3 μ 1 2 + σ 4 μ 1 4 = 0 , $$ {\sigma }_2+{\sigma }_3{\mu }_1^2+{\sigma }_4{\mu }_1^4=0, $$(88)provided that 2 σ 3 + 4 σ 4 μ 1 2 > 0 $ 2{\sigma }_3+4{\sigma }_4{\mu }_1^2>0$ to ensure the validity of constructing quasi-soliton wave. Making use of these results, the coupled equations (2) and (3) has chirped dark quasi-soliton solution presented as q ( x , t ) = μ 1 tanh [ Λ ( x - ν t ) ] 1 + μ 2 sech 2 [ Λ ( x - ν t ) ] e i ( ϕ ( ξ ) - ω t ) , $$ q\left(x,t\right)=\frac{{\mu }_1\mathrm{tanh}\left[\mathrm{\Lambda }\left(x-{\nu t}\right)\right]}{\sqrt{1+{\mu }_2{\mathrm{sech}}^2\left[\mathrm{\Lambda }\left(x-{\nu t}\right)\right]}}{\mathrm{e}}^{\mathrm{i}\left(\phi \left(\xi \right)-{\omega t}\right)}, $$(89) r ( x , t ) = γ q ( x , t ) . $$ r\left(x,t\right)={\gamma q}\left(x,t\right). $$(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 coupled-KE 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 a1 = a2 = 1, γ = α1 = α2 = ρ1 = m1 = p1 = 0.5, n1 = −0.5, s1 = 1.5, A6 = 4. Based on the change in the value of A4, 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 kink-dark soliton with A4 = 8 while Figure 1b exhibits kink wave with A4 = 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 A6 = −4 and with different values of A4 and l1. The first soliton form represents bright soliton wave as depicted in Figure 2a when A4 = 2; l1 = −0.5, −0.3, −0.1, the second soliton form describes soliton wave having W shape as shown in Figure 2b when A4 = 4; l1 = 0.3, 0.6, 1 and the third wave form is dark soliton as presented in Figure 2c when A4 = −2; l1 = −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 A4 = 8, ω = 1, β = 0.5. The bright soliton in Figure 3a is plotted with a1 = 1, the W-shaped soliton in 3b is plotted with a1 = −2.5 and the dark soliton in 3c is plotted with a1 = −1. In Figure 4, the graph illustrates anti-kink soliton characterizing solutions (59) and (60) for the values of parameters a1 = a2 = b1 = c1 = 1, γ = α1 = α2 = ρ1 = η1 = η3 = d1 = 0.5, A6 = 4, A4 = 8 while Figure 5 depicts dark soliton profile that represents solutions (65) and (66) where A6 = −4. Moreover, we observe that Figure 6 presents three solitonic structures describing solutions (72) and (73) for the values of parameters a1 = a2 = η3 = 1, γ = α1 = α2 = ρ1 = 0.5, A4 = 8, A6 = 4. The first structure is kink-dark soliton as displayed in Figure 6a with η2 = 1, η4 = 0.1 and η1 = 0.1, 0.3, 0.5. The second structure is kink soliton as plotted in Figure 6b with η1 = 1, η4 = 0.9 and η2 = 0.1, 0.4, 0.7. The third structure is anti-kink soliton as presented in Figure 2c with η1 = 1, η2 = 0.9 and η4 = 0.1, 0.4, 0.7. Obviously, one can see that Figure 7 demonstrates three wave forms which are bright, W-shaped and dark solitons describing solutions (78) and (79) with same values of parameters as in Figure 6 besides η1 = 1, A6 = −4. The bright soliton is shown in Figure 7a with η2 = 1, η4 = 0.1, 0.4, 0.8; the W-shaped soliton is shown in Figure 7b with η2 = −1.2, η4 = 0.1, 0.5, 1 and the dark soliton is shown in Figure 7c with η4 = 1, η2 = 0.1, 0.4, 0.8. The special case of chirped bright quasi-soliton solution (84) and (85) is depicted in Figure 8 with same values of parameters as in Figure 2 and ω = 1, β = n1 = 0.5. Further to this, the chirped dark quasi-soliton solution (89) and (90) is delineated in Figure 9 with same values of parameters as in Figure 8 and μ = 1, a1 = −0.5, −1.5, −2.5.

thumbnail Figure 1

Soliton intensity for q(x; t) and r(x; t) given in (41) and (42) along with chirping profile.

thumbnail Figure 2

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

thumbnail Figure 3

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

thumbnail Figure 4

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

thumbnail Figure 5

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

thumbnail Figure 6

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

thumbnail Figure 7

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

thumbnail Figure 8

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

thumbnail 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 18, 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 quasi-soliton 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 self-phase 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, W-shaped, kink, anti-kink and Kink-dark 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 quasi-soliton 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. Al-Ghafri and M. Sankar gratefully acknowledge support provided by University of Technology and Applied Sciences, Ibri, Oman through the Internal Research Funding Program, grant number 01-IRFP-IBRI-2023.

References

  1. Agrawal G.P. (2000) Nonlinear fiber optics, Nonlinear Science at the Dawn of the 21st Century, Springer, Berlin, Heidelberg, pp. 195–211. [NASA ADS] [CrossRef] [Google Scholar]
  2. Katzir A. (2012) Lasers and optical fibers in medicine, in: Physical Techniques in Biology and Medicine, Elsevier Science. [Google Scholar]
  3. Minakuchi S., Takeda N. (2013) Recent advancement in optical fiber sensing for aerospace composite structures, Photon. Sens. 3, 345–354. [NASA ADS] [CrossRef] [Google Scholar]
  4. Schartner E.P., Tsiminis G., François A., Kostecki R., Warren-Smith S.C., Nguyen L.V., Heng S., Reynolds T., Klantsataya E., Rowland K.J., et al. (2015) Taming the light in microstructured optical fibers for sensing, Int. J. Appl. Glass Sci. 6, 229–239. [CrossRef] [Google Scholar]
  5. De Angelis C. (2021) Nonlinear optics, Front. Photon. 1, 628215. [CrossRef] [Google Scholar]
  6. Pal B.P. (2005) Guided wave optical components and devices: basics, technology, and applications, Indian Institute of Technology, Delhi, India. [Google Scholar]
  7. Zhongwei T., Chao L. (2020) Optical fiber communication technology: Present status and prospect, Strategic Study CAE 22, 100–107. [CrossRef] [Google Scholar]
  8. Kaminow I.P., Li T. (2002) Optical fiber telecommunications IV-B: systems and impairments, in: Optics and Photonics, Elsevier Science. [Google Scholar]
  9. Utzinger U., Richards-Kortum R.R. (2003) Fiber optic probes for biomedical optical spectroscopy, J. Biomed. Opt. 8, 121–147. [NASA ADS] [CrossRef] [Google Scholar]
  10. Yariv A., Yeh P. (2007) Photonics: optical electronics in modern communications, Oxford University Press. [Google Scholar]
  11. Bufetov I.A., Melkumov M.A., Firstov S.V., Riumkin K.E., Shubin A.V., Khopin V.F., Guryanov A.N., Dianov E.M. (2014) Bi-doped optical fibers and fiber lasers. IEEE J. Sel. Top. Quantum Electron. 20, 111–125. [NASA ADS] [CrossRef] [Google Scholar]
  12. Rajan G. (2017) Optical fiber sensors: advanced techniques and applications, CRC Press. [CrossRef] [Google Scholar]
  13. Addanki S., Amiri I.S., Yupapin P. (2018) Review of optical fibers-introduction and applications in fiber lasers, Results Phys. 10, 743–750. [NASA ADS] [CrossRef] [Google Scholar]
  14. Biswas A., Ekici M., Sonmezoglu A., Belic M.R. (2019) Optical solitons in fiber Bragg gratings with dispersive reflectivity for quadratic–cubic nonlinearity by extended trial function method, Optik 185, 50–56. [NASA ADS] [CrossRef] [Google Scholar]
  15. Zayed E.M., Alngar M.E., Biswas A., Triki H., Yıldırım Y., Alshomrani A.S. (2020) Chirped and chirp-free optical solitons in fiber Bragg gratings with dispersive reflectivity having quadratic-cubic nonlinearity by sub-ODE approach, Optik 203, 163993. [NASA ADS] [CrossRef] [Google Scholar]
  16. Zayed E., Alngar M., Biswas A., Ekici M., Alzahrani A., Belic M. (2020) Chirped and chirp-free optical solitons in fiber Bragg gratings with Kudryashov’s model in presence of dispersive reflectivity, J. Commun. Technol. Electron. 65, 1267–1287. [Google Scholar]
  17. Yıldırım Y., Biswas A., Guggilla P., Khan S., Alshehri H.M., Belic M.R. (2021) Optical solitons in fibre Bragg gratings with third-and fourth-order dispersive reflectivities, Ukr. J. Phys. Opt. 22, 239–254. [CrossRef] [Google Scholar]
  18. Malik S., Kumar S., Biswas A., Ekici M., Dakova A., Alzahrani A.K., Belic M.R. (2021) Optical solitons and bifurcation analysis in fiber Bragg gratings with Lie symmetry and Kudryashov’s approach, Nonlinear Dyn. 105, 735–751. [Google Scholar]
  19. Yıldırım Y., Biswas A., Khan S., Guggilla P., Alzahrani A.K., Belic M.R. (2021) Optical solitons in fiber Bragg gratings with dispersive reflectivity by sine-Gordon equation approach, Optik 237, 166684. [CrossRef] [Google Scholar]
  20. Al-Ghafri K.S., Sankar M., Krishnan E.V., Khan S., Biswas A. (2023) Chirped gap solitons in fiber Bragg gratings with polynomial law of nonlinear refractive index, Journal of the European Optical Society 19, .30 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  21. Zhong Y., Triki H., Zhou Q. (2023) Analytical and numerical study of chirped optical solitons in a spatially inhomogeneous polynomial law fiber with parity-time symmetry potential., Commun. Theoret. Phys. 75, 025003. [NASA ADS] [CrossRef] [Google Scholar]
  22. Zhou Q., Triki H., Xu J., Zeng Z., Liu W., Biswas A. (2022) Perturbation of chirped localized waves in a dual-power law nonlinear medium, Chaos Solitons Fractals 160, 112198. [NASA ADS] [CrossRef] [Google Scholar]
  23. Zhou Q., Zhong Y., Triki H., Sun Y., Xu S., Liu W., Biswas A. (2022) Chirped bright and kink solitons in nonlinear optical fibers with weak nonlocality and cubic-quantic-septic nonlinearity, Chin. Phys. Lett. 39, 044202. [NASA ADS] [CrossRef] [Google Scholar]
  24. Zhou Q. (2022) Influence of parameters of optical fibers on optical soliton interactions, Chin. Phys. Lett. 39, 010501. [NASA ADS] [CrossRef] [Google Scholar]
  25. Zhou Q., Huang Z., Sun Y., Triki H., Liu W., Biswas A. (2023) Collision dynamics of three-solitons in an optical communication system with third-order dispersion and nonlinearity, Nonlin. Dynamics 111, 5757–5765. [CrossRef] [Google Scholar]
  26. Sun Y., Hu Z., Triki H., Mirzazadeh M., Liu W., Biswas A., Zhou Q. (2023) Analytical study of three-soliton interactions with different phases in nonlinear optics, Nonlin. Dyn. 111, 18391–18400. [CrossRef] [Google Scholar]
  27. Zhou Q., Sun Y., Triki H., Zhong Y., Zeng Z., Mirzazadeh M. (2022) Study on propagation properties of one-soliton in a multimode fiber with higher-order effects, Results Phys., 41, 105898. [NASA ADS] [CrossRef] [Google Scholar]
  28. Kudryashov N.A. (2019) A generalized model for description of propagation pulses in optical fiber, Optik 189, 42–52. [NASA ADS] [CrossRef] [Google Scholar]
  29. Biswas A., Sonmezoglu A., Ekici M., Alshomrani A.S., Belic M.R. (2019) Optical solitons with Kudryashov’s equation by F-expansion, Optik 199, 163338. [NASA ADS] [CrossRef] [Google Scholar]
  30. Biswas A., Vega-Guzmán J., Ekici M., Zhou Q., Triki H., Alshomrani A.S., Belic M.R. (2020) Optical solitons and conservation laws of Kudryashov’s equation using undetermined coefficients, Optik 202, 163417. [CrossRef] [Google Scholar]
  31. Kumar S., Malik S., Biswas A., Zhou Q., Moraru L., Alzahrani A., Belic M. (2020) Optical solitons with Kudryashov’s equation by Lie symmetry analysis, Phys. Wave Phenom. 28, 299–304. [NASA ADS] [CrossRef] [Google Scholar]
  32. Arnous A.H., Biswas A., Ekici M., Alzahrani A.K., Belic M.R. (2021) Optical solitons and conservation laws of Kudryashov’s equation with improved modified extended tanh-function, Optik 225, 165406. [NASA ADS] [CrossRef] [Google Scholar]
  33. Zayed E.M., Alngar M.E. (2021) Optical soliton solutions for the generalized Kudryashov equation of propagation pulse in optical fiber with power nonlinearities by three integration algorithms, Math. Methods Appl. Sci. 44, 315–324. [NASA ADS] [CrossRef] [Google Scholar]
  34. Hu X., Yin Z. (2022) A study of the pulse propagation with a generalized Kudryashov equation, Chaos, Solitons Fractals 161, 112379. [NASA ADS] [CrossRef] [Google Scholar]
  35. Khuri S., Wazwaz A.-M. (2023) Optical solitons and traveling wave solutions to Kudryashov’s equation, Optik 279, 170741. [NASA ADS] [CrossRef] [Google Scholar]
  36. Kumar S., Niwas M. (2023) Optical soliton solutions and dynamical behaviours of Kudryashov’s equation employing efficient integrating approach, Pramana 97, 98. [CrossRef] [Google Scholar]
  37. Kudryashov N.A., Antonova E.V. (2020) Solitary waves of equation for propagation pulse with power nonlinearities, Optik 217, 164881. [NASA ADS] [CrossRef] [Google Scholar]
  38. Kudryashov N.A. (2020) Mathematical model of propagation pulse in optical fiber with power nonlinearities, Optik 212, 164750. [NASA ADS] [CrossRef] [Google Scholar]
  39. Kudryashov N.A. (2020) Optical solitons of mathematical model with arbitrary refractive index, Optik 224, 165391. [NASA ADS] [CrossRef] [Google Scholar]
  40. Zayed E., Alurrfi K. (2016) New extended auxiliary equation method and its applications to nonlinear Schrödinger-type equations, Optik 127, 9131–9151. [NASA ADS] [CrossRef] [Google Scholar]

All Figures

thumbnail Figure 1

Soliton intensity for q(x; t) and r(x; t) given in (41) and (42) along with chirping profile.

In the text
thumbnail Figure 2

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

In the text
thumbnail Figure 3

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

In the text
thumbnail Figure 4

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

In the text
thumbnail Figure 5

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

In the text
thumbnail Figure 6

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

In the text
thumbnail Figure 7

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

In the text
thumbnail Figure 8

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

In the text
thumbnail 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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