The existence of soliton in the filed of optical fibers has made a wide improvement in various sectors of our life such as communications, medical, aerospace and many others [1–3]. In communications, for example, soliton operates as one of the effective carriers that transmit huge information through optical fibers over transcontinental and transoceanic distances [4–8]. Basically, soliton in fiber-optic communication systems is known as optical soliton which denotes a light pulse propagating through a nonlinear optical medium without any change in its shape and velocity [9–11]. The emergence of optical soliton is based on controllable interaction of dispersion and nonlinearity of the pulse propagation. The investigation of optical soliton behaviors can be done through discussing a model of the nonlinear Schrödinger equation (NLSE) or its generalized forms [12–16].

During propagation scenario within an optical fiber, soliton is found to be badly affected by the low count of chromatic dispersion (CD) and hence the data transmission is obfuscated. This problem can be handled via applying a new technology, namely, dispersion compensating fiber. One of the perfect candidates for this role is Bragg gratings (BGs) because of their low loss, small footprint, and low optical nonlinearity. Practically, BGs provides induced dispersion to replenish the low count of CD and sufficiently ensure the smooth formation of solitons which are transmitted along fibers for intercontinental distances. Based on the physical nature of fiber BGs medium, the self-phase modulation arises from the nonlinear refractive index structures which has different types such as Kerr law, quadratic-cubic law, parabolic law, polynomial law, parabolic-nonlocal combo law and many others. This diversity of forms of nonlinear refractive index leads to create distinct forms of NLSE.

Various studies in literature are implemented in the area of fiber BGs under the influences of different types of nonlinearity to examine the behaviors of optical solitons by making use of powerful integration schemes. For instance, Biswas et al. [17] scrutinized coupled NLSE in fiber BGs with parabolic form of nonlinearity. They extracted three forms of solitons called bright, dark and singular solitons. With the aid of modified simple equation, dark and singular optical solitons to fiber Bragg gratings with Kerr law are obtained by a group of authors [18]. The latter model is studied as well in presence of the four-wave mixing terms [19]. As a result, chirped and chirp-free bright, dark and singular solitons are revealed. Moreover, solitons in fiber BGs with five forms of nonlinear refractive index are investigated using the sine-Gordon equation method [20]. Distinct structures of solitons including bright, dark, singular and combo singular solitons are retrieved. For more details about the previous studies that discussed optical solitons in fiber BGs, the reader is referred to the references [21–35].

The recent experimental studies declare that optical solitons with nonlinear chirping have effective role in engineering applications such as the design of fiber-optic amplifiers, spread spectrum communications, photonic and optoelectronic devices. Hence, a lot of intensive theoretical works are directed to the investigation of chirped solitons in fiber-optic media. Many experts have dealt with various mathematical models to derive miscellaneous types of chirped solitons by means of several analytical techniques. Some of obtained soliton structures include kink, dark and bright solitons [36–39]; chirped self-similar bright and kink solitons [40]; dark-dark and bright-bright soliton pairs [41]; chirped self-similar gray and kink waves [42], see also [43–46]. Our present work focuses on deriving chirped optical soliton solutions in fiber BGs. The nonlinear chirping associated to each soliton solution is also created.

The most substantial purpose of this study is to inquire into the dimensionless form of the coupled NLSE in fiber BGs having polynomial law of nonlinearity given by [47, 48](1)$$i{q}_t+a_1{r}_{\mathrm{xx}}+\left(b_1\right|q{|}^2+c_1\left|r{|}^2\right)q+\left(d_1\right|q{|}^4+f_1\left|q{|}^2\right|r{|}^2+g_1\left|r{|}^4\right)q+\left(l_1\right|q{|}^6+m_1\left|q{|}^4\right|r{|}^2+n_1\left|q{|}^2\right|r{|}^4+p_1\left|r{|}^6\right)q+i{h}_1{q}_x+k_{1}r=0,$$(2)$$i{r}_t+a_2{q}_{\mathrm{xx}}+\left(b_2\right|r{|}^2+c_2\left|q{|}^2\right)r+\left(d_2\right|r{|}^4+f_2\left|r{|}^2\right|q{|}^2+g_2\left|q{|}^4\right)r+\left(l_2\right|r{|}^6+m_2\left|r{|}^4\right|q{|}^2+n_2\left|r{|}^2\right|q{|}^4+p_2\left|q{|}^6\right)r+i{h}_2{r}_x+k_{2}q=0,$$where the functions q(x, t) and r(x, t) indicate forward and backward propagating waves, respectively, while a_j for j = 1, 2 denote the coefficients of dispersive reflectivity. The terms having b_j account for the coefficients of self-phase modulation (SPM) whereas the terms with c_j represent the cross-phase modulation (XPM) for cubic nonlinearity portion. Regarding quintic nonlinear part, d_j are the coefficients of SPM while f_j and g_j are the coefficients of XPM. For septic nonlinearity, l_j are the coefficients of SPM while m_j, n_j and p_j are the coefficients of XPM. Finally, h_j stand for inter-modal dispersion and k_j represent detuning parameters. All of the coefficients are real valued constants and $i=\sqrt{-1}$.

This work is essentially devoted to investigating chirped optical solitons in fiber BG with polynomial law of nonlinear refractive index. The coupled NLSE (1) and (2) is dealt with analytic strategy and reduced to an integrable form under the phase-matching condition. Then, soliton solutions are created by implementing two forms for the method of undetermined coefficients. The corresponding chirp to each soliton solution is also derived as a nonlinear function in terms of the reciprocal of intensity. The physical interpretations of constructed optical solitons are displayed.

Our aim now is to analyze the system of coupled NLSE (1) and (2) and to reduce it to an integrable form. To achieve this process, a complex variable transformation is assumed as(3)$$q(x,t)=u_1\left(\tau \right)e^{i\left(\mathrm{{\rm Y}}\right(\tau )-\omega t)},$$(4)$$r(x,t)=u_2\left(\tau \right)e^{i\left(\mathrm{{\rm Y}}\right(\tau )-\omega t)},$$where τ = x − νt while ω and ν are real constants indicating the wave number and the soliton velocity. The functions u₁(τ) and u₂(τ) stand for the amplitudes of the solitons whereas Υ(τ) represents the nonlinear phase shift. The corresponding chirp is identified as $\delta \omega (x,t)=-\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left[\mathrm{{\rm Y}}\right(\tau )-\omega t]=-\mathrm{{\rm Y}}\mathrm{\text{'}}\left(\tau \right)$. Substituting (3) and (4) into the coupled system (1) and (2) and then separating the imaginary and real components, one can arrive at(5)$${a_1{u}_2^{″}-a_1{u}_2\mathrm{{\rm Y}}\mathrm{\text{'}}}^2+(\nu -h_1)u_1\mathrm{{\rm Y}}\mathrm{\text{'}}+\omega u_1+k_1{u}_2+b_1{u}_1^3+c_1{u}_1{u}_2^2+d_1{u}_1^5+f_1{u}_1^3{u}_2^2+g_1{u}_1{u}_2^4+l_1{u}_1^7+m_1{u}_1^5{u}_2^2+n_1{u}_1^3{u}_2^4+p_1{u}_1{u}_2^6=0,$$(6)$${a_2{u}_1^{″}-a_2{u}_1\mathrm{{\rm Y}}\mathrm{\text{'}}}^2+(\nu -h_2)u_2\mathrm{{\rm Y}}\mathrm{\text{'}}+\omega u_2+k_2{u}_1+b_2{u}_2^3+c_2{u}_2{u}_1^2+d_2{u}_2^5+f_2{u}_2^3{u}_1^2+g_2{u}_2{u}_1^4+l_2{u}_2^7+m_2{u}_2^5{u}_1^2+n_2{u}_2^3{u}_1^4+p_2{u}_2{u}_1^6=0,$$and(7)$$(h_1-\nu )u_1^{\text{'}}+a_1(u_2\mathrm{{\rm Y}}\mathrm{″}+2u_2\mathrm{{\rm Y}}\mathrm{\text{'}})=0,$$(8)$$(h_2-\nu )u_2^{\text{'}}+a_2(u_1\mathrm{{\rm Y}}\mathrm{″}+2u_1\mathrm{{\rm Y}}\mathrm{\text{'}})=0.$$

To manipulate the equations obtained above, the following relation is proposed as(9)$$u_2=\rho u_1,$$where ρ ≠ 1 is a real constant. Subsequently, the set of equations (5)–(8) change into(10)$${\rho a_1{u}_1^{″}+[\omega +\rho k_1+(\nu -h_1)\mathrm{{\rm Y}}\mathrm{\text{'}}-\rho a_1\mathrm{{\rm Y}}\mathrm{\text{'}}}^2]u_1+(b_1+{\rho }^2{c}_1)u_1^3+(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)u_1^5+(l_1+{\rho }^2{m}_1+{\rho }^4{n}_1+{\rho }^6{p}_1)u_1^7=0,$$(11)$${a_2{u}_1^{″}+[\rho \omega +k_2+\rho (\nu -h_2)\mathrm{{\rm Y}}\mathrm{\text{'}}-a_2\mathrm{{\rm Y}}\mathrm{\text{'}}}^2]u_1+({\rho }^3{b}_2+\rho c_2)u_1^3+({\rho }^5{d}_2+{\rho }^3{f}_2+\rho g_2)u_1^5+({\rho }^7{l}_2+{\rho }^5{m}_2+{\rho }^3{n}_2+\rho p_2)u_1^7=0,$$and(12)$$(h_1-\nu )u_1^{\text{'}}+\rho a_1(u_1\mathrm{{\rm Y}}\mathrm{″}+2u_1\mathrm{{\rm Y}}\mathrm{\text{'}})=0,$$(13)$$\rho (h_2-\nu )u_1^{\text{'}}+a_2(u_1\mathrm{{\rm Y}}\mathrm{″}+2u_1\mathrm{{\rm Y}}\mathrm{\text{'}})=0.$$

From the integrability of equations (12) and (13), one can obtain(14)$$\mathrm{{\rm Y}}\mathrm{\text{'}}=\frac{\nu -h_1}{2\rho a_1}+\frac{{\delta }_1{u}_1^{-2}}{\rho a_1},$$(15)$$\mathrm{{\rm Y}}\mathrm{\text{'}}=\frac{\rho (\nu -h_2)}{2a_2}+\frac{{\delta }_2{u}_1^{-2}}{a_2},$$where δ₁ and δ₂ are the integration constants. Since equations (14) and (15) are equivalent, they give rise the constraint condition of the form(16)$$(a_2-{\rho }^2{a}_1)\nu u_1^2-(h_1{a}_2-h_2{\rho }^2{a}_1)u_1^2+2({\delta }_1{a}_2-{\delta }_2\rho a_1)=0.$$

To reduce the level of complexity in the system of equations (14) and (15), we assume that(17)$${\delta }_2\rho a_1={\delta }_1{a}_2,$$and, as a consequence, equation (16) collapses to the expression for the velocity of the soliton as(18)$$\nu =\frac{h_1{a}_2-h_2{\rho }^2{a}_1}{a_2-{\rho }^2{a}_1}.$$

As a result, the chirp can be written as(19)$$\delta \omega (x,t)=-\left[\frac{\nu -h_1}{2\rho a_1}+\frac{{\delta }_1{u}_1^{-2}}{\rho a_1}\right].$$

Plugging (14) and (15) into equations (10) and (11), respectively, we find(20)$$\rho a_1{u}_1^{″}-\frac{{\delta }_1^2}{\rho a_1}{u}_1^{-3}+\left[\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right]u_1+(b_1+{\rho }^2{c}_1)u_1^3+(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)u_1^5+(l_1+{\rho }^2{m}_1+{\rho }^4{n}_1+{\rho }^6{p}_1)u_1^7=0,$$(21)$$a_2{u}_1^{″}-\frac{{\delta }_1^2{a}_2}{{\rho }^2{a}_1^2}{u}_1^{-3}+\left[\rho \omega +k_2+\frac{{\rho }^2(\nu -h_2{)}^2}{4a_2}\right]u_1+({\rho }^3{b}_2+\rho c_2)u_1^3+({\rho }^5{d}_2+{\rho }^3{f}_2+\rho g_2)u_1^5+({\rho }^7{l}_2+{\rho }^5{m}_2+{\rho }^3{n}_2+\rho p_2)u_1^7=0.$$

Integrating the last equations, this leads to(22)$$\rho a_1{\mathrm{u\text{'}}}_1^2-\frac{{\delta }_1^2}{\rho a_1}{u}_1^{-2}+\left[\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right]u_1^2+\frac{1}{2}(b_1+{\rho }^2{c}_1)u_1^4+\frac{1}{3}(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)u_1^6+\frac{1}{4}(l_1+{\rho }^2{m}_1+{\rho }^4{n}_1+{\rho }^6{p}_1)u_1^8+2{\mu }_1=0,$$(23)$$a_2{\mathrm{u\text{'}}}_1^2-\frac{{\delta }_1^2{a}_2}{{\rho }^2{a}_1^2}{u}_1^{-2}+\left[\rho \omega +k_2+\frac{{\rho }^2(\nu -h_2{)}^2}{4a_2}\right]u_1^2+\frac{1}{2}({\rho }^3{b}_2+\rho c_2)u_1^4+\frac{1}{3}({\rho }^5{d}_2+{\rho }^3{f}_2+\rho g_2)u_1^6+\frac{1}{4}({\rho }^7{l}_2+{\rho }^5{m}_2+{\rho }^3{n}_2+\rho p_2)u_1^8+2{\mu }_2=0,$$where μ₁ and μ₂ are the constants of integration. Equations (22) and (23) are equivalent under the following conditions presented as(24)$$a_2=\rho a_1,$$(25)$$\rho \omega +k_2+\frac{{\rho }^2(\nu -h_2{)}^2}{4a_2}=\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1},$$(26)$${\rho }^3{b}_2+\rho c_2=b_1+{\rho }^2{c}_1,$$(27)$${\rho }^5{d}_2+{\rho }^3{f}_2+\rho g_2=d_1+{\rho }^2{f}_1+{\rho }^4{g}_1,$$(28)$${\rho }^7{l}_2+{\rho }^5{m}_2+{\rho }^3{n}_2+\rho p_2=l_1+{\rho }^2{m}_1+{\rho }^4{n}_1+{\rho }^6{p}_1,$$(29)$${\mu }_2={\mu }_1.$$

As both equations (22) and (23) are equivalent, the rest of procedure in this work are performed through tackling equation (22). Let us introduce the variable transformation given by(30)$$u_1^2=F,$$which enables us to convert equation (22) into(31)$${\rho a_{1}F\mathrm{\prime }}^2-\frac{4{\delta }_1^2}{\rho a_1}+8{\mu }_{1}F+4\left[\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right]F^2+2(b_1+{\rho }^2{c}_1)F^3+\frac{4}{3}(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)F^4+(l_1+{\rho }^2{m}_1+{\rho }^4{n}_1+{\rho }^6{p}_1)F^5=0.$$

Equation (31) can be rearranged to have the form(32)$$\rho a_1\mathrm{F″}+4{\mu }_1+4\left[\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right]F+3(b_1+{\rho }^2{c}_1)F^2+\frac{8}{3}(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)F^3+\frac{5}{2}(l_1+{\rho }^2{m}_1+{\rho }^4{n}_1+{\rho }^6{p}_1)F^4=0.$$

To derive the chirped soliton solutions, the method of undetermined coefficients with two forms having the hyperbolic secant and tangent functions is employed to equation (32). Then, the obtained solutions are inserted into the relation (30) together with (3) to arrive at the chirped optical solitons for the coupled system (1) and (2).

We express that the solution of equation (32) has the form(33)$$F\left(\tau \right)={\alpha }_0+{\alpha }_1\tan \mathrm{h}\left({\alpha }_3\tau \right)+{\alpha }_2{\tan \mathrm{h}}^2\left({\alpha }_3\tau \right),$$where α₀, α₁, α₂ and α₃ are constants to be identified. Substituting the ansatz (33) into equation (32) and then equating all coefficients of all powers of tan h(α₃τ) to zero, one can obtain a system of algebraic equations that determines the value of constants α₀, α₁, α₂ and α₃. Solving this system brings about the following sets of solutions.

(34)$${\alpha }_0=\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{3(b_1+{\rho }^2{c}_1)},$$(35)$${\alpha }_1=0,$$(36)$${\alpha }_2=-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)},$$(37)$${\alpha }_3=\sqrt{\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}},$$under the constraint conditions(38)$$d_1+{\rho }^2{f}_1+{\rho }^4{g}_1=0,$$(39)$$l_1+{\rho }^2{m}_1+{\rho }^4{n}_1+{\rho }^6{p}_1=0.$$

Substituting (34)–(37) into (33) and using the relations (30) and (9) along with (3) and (4), we arrive at an exact solution in the form of chirped bright soliton for the coupled system (2) and (3) as(40)$$\begin{array}{ll}& q(x,t)={\left[\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{3(b_1+{\rho }^2{c}_1)}\left\{1-3{\tan \mathrm{h}}^2\left(\sqrt{\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right\}\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\\ & r(x,t)=\rho {\left[\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{3(b_1+{\rho }^2{c}_1)}\left\{1-3{\tan \mathrm{h}}^2\left(\sqrt{\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right\}\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\end{array}$$where $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)\left(\rho a_1\right)>0$. The associated chirp is presented as(41)$$\begin{array}{c}\delta \omega (x,t)=-\left[\frac{\nu -h_1}{2\rho a_1}+\frac{{\delta }_1}{\rho a_1}\left\{-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{3(b_1+{\rho }^2{c}_1)}\right.\right.\\ \left.{\left.\times \left[1-3{\tan \mathrm{h}}^2\left(\sqrt{\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right]\right\}}^{-1}\right].\end{array}$$

(42)$${\alpha }_0=-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)},$$(43)$${\alpha }_1=0,$$(44)$${\alpha }_2=\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)},$$(45)$${\alpha }_3=\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}},$$under the constraint conditions (38) and (39). Substituting these outcomes into (33) and using the relations (30) and (9) together with (3) and (4), we obtain the chirped bright soliton for the coupled system (2) and (3) given by(46)$$\begin{array}{ll}& q(x,t)=\sqrt{-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}}\sec \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)e^{i\left(\phi \right(\tau )-\omega t)},\\ & r(x,t)=\rho \sqrt{-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}}\sec \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)e^{i\left(\phi \right(\tau )-\omega t)},\end{array}$$where $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)\left(\rho a_1\right)<0$ and $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)(b_1+{\rho }^2{c}_1)<0$. The chirp expression is addressed as(47)$$\delta \omega (x,t)=-\left[\frac{\nu -h_1}{2\rho a_1}-\frac{{\delta }_1}{\rho a_1}{\left\{\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}\sec {\mathrm{h}}^2\left(\sqrt{-\frac{\omega +\rho {\mathrm{k}}_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right\}}^{-1}\right].$$

(48)$${\alpha }_0=-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)},$$(49)$${\alpha }_1=-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)},$$(50)$${\alpha }_2=0,$$(51)$${\alpha }_3=\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}},$$under the constraint conditions (39) and(52)$$16\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)=3(b_1+{\rho }^2{c}_1{)}^2.$$

Exploiting (33)–(51) with (33) and using the relations (30) and (9) together with (3) and (4), we present the chirped dark soliton pair for the coupled system (2) and (3) as(53)$$\begin{array}{ll}& q(x,t)={\left[-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}\left\{1+\tan \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right\}\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\\ & r(x,t)=\rho {\left[-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}\left\{1+\tan \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right\}\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\end{array}$$where $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)\left(\rho a_1\right)<0$ and $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)(b_1+{\rho }^2{c}_1)<0$. The corresponding chirping is introduced as(54)$$\delta \omega (x,t)=-\left[\frac{\nu -h_1}{2\rho a_1}-\frac{{\delta }_1}{\rho a_1}{\left\{\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}\left[1+\tan \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right]\right\}}^{-1}\right].$$

(55)$${\alpha }_0=-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)},$$(56)$${\alpha }_1=\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)},$$(57)$${\alpha }_2=0,$$(58)$${\alpha }_3=\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}},$$under the constraint conditions (39) and (52). Plugging these findings into (33) and using the relations (30) and (9) together with (3) and (4), we secure the chirped dark soliton pair for the coupled system (2) and (3) as(59)$$\begin{array}{ll}& q(x,t)={\left[-\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}\left\{1-\tan \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right\}\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\\ & r(x,t)=\rho {\left[-\frac{2\left(\omega +\mathrm{\rho }{k}_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}\left\{1-\tan \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right\}\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\end{array}$$where $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)\left(\rho a_1\right)<0$ and $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)(b_1+{\rho }^2{c}_1)<0$. The associated chirp is caught in the form(60)$$\delta \omega (x,t)=-\left[\frac{\nu -h_1}{2\rho a_1}-\frac{{\delta }_1}{\rho a_1}{\left\{\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{(b_1+{\rho }^2{c}_1)}\left[1-\tan \mathrm{h}\left(\sqrt{-\frac{\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}}{\rho a_1}}\tau \right)\right]\right\}}^{-1}\right].$$

(61)$${\alpha }_0=0,$$(62)$${\alpha }_1=\sqrt{-\frac{3\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{2(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)}},$$(63)$${\alpha }_2=0,$$(64)$${\alpha }_3=\sqrt{\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{\rho a_1}},$$under the constraint conditions (39) and(65)$$b_1+{\rho }^2{c}_1=0.$$

Employing (61)–(64) with (33) and using the relations (30) and (9) together with (3) and (4), we acquire the chirped dark soliton solution for the coupled system (2) and (3) as(66)$$\begin{array}{ll}& q(x,t)={\left[\sqrt{-\frac{3\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{2(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)}}\tan \mathrm{h}\left(\sqrt{\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{\rho a_1}}\tau \right)\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\\ & r(x,t)=\rho {\left[\sqrt{-\frac{3\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{2(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)}}\tan \mathrm{h}\left(\sqrt{\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{\rho a_1}}\tau \right)\right]}^{\frac{1}{2}}{e}^{i\left(\phi \right(\tau )-\omega t)},\end{array}$$where $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)\left(\rho a_1\right)>0$ and $\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)<0$. The chirping is procured as(67)$$\delta \omega (x,t)=-\left[\frac{\nu -h_1}{2\rho a_1}+\frac{{\delta }_1}{\rho a_1}{\left\{\sqrt{-\frac{3\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{2(d_1+{\rho }^2{f}_1+{\rho }^4{g}_1)}}\tan \mathrm{h}\left(\sqrt{\frac{2\left(\omega +\rho k_1+\frac{(\nu -h_1{)}^2}{4\rho a_1}\right)}{\rho a_1}}\tau \right)\right\}}^{-1}\right].$$

We consider that equation (32) has an exact soliton solution identified as(68)$$F\left(\tau \right)={\beta }_0+\frac{{\beta }_1\sec \mathrm{h}\left({\beta }_3\tau \right)}{1+\sec \mathrm{h}\left({\beta }_3\tau \right)}+\frac{{\beta }_2\sec {\mathrm{h}}^2\left({\beta }_3\tau \right)}{1+\sec {\mathrm{h}}^2\left({\beta }_3\tau \right)},$$where β₀, β₁, β₂ and β₃ are constants to be determined. Applyiny ansatz (68) to equation (32) gives rise to a polynomial in sec h(β₃τ) of various powers. Equating each coefficient in this polynomial to zero yields a system of algebraic equations that induces the following sets of solutions.