# Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quinic Nonlinearity
## 2 The Model and Its Linear Spectrum  
To derive the mathematical model for the system, we start from [PERSON]'s equations and following the steps for the derivation of GMTM [9], the coupled mode equations [49], and the approaches for the inclusion of dispersive reflectivity and cubic-quintic nonlinearity described, respectively, in References [36; 40] the following system of normalized partial differential equations can be derived:  
\[iu_{1t}+iu_{1x}+\bigg{(}\frac{1}{2}|u_{1}|^{2}+|v_{1}|^{2}\bigg{)}u_{1}-\eta \bigg{(}\frac{1}{4}|u_{1}|^{4}+\frac{3}{2}|u_{1}|^{2}|v_{1}|^{2}+\frac{3}{4}|v _{1}|^{4}\bigg{)}u_{1}+v_{1}+\lambda u_{2}+mv_{1 xx}=0, \tag{1}\] \[iv_{1t}-iv_{1x}+\bigg{(}\frac{1}{2}|v_{1}|^{2}+|u_{1}|^{2}\bigg{)} v_{1}-\eta\bigg{(}\frac{1}{4}|v_{1}|^{4}+\frac{3}{2}|v_{1}|^{2}|u_{1}|^{2}+\frac{3}{4}|u _{1}|^{4}\bigg{)}v_{1}+u_{1}+\lambda v_{2}+mu_{1 xx}=0,\] \[iu_{2t}+iu_{2x}+\bigg{(}\frac{1}{2}|u_{2}|^{2}+|v_{2}|^{2}\bigg{)} u_{2}-\eta\bigg{(}\frac{1}{4}|u_{2}|^{4}+\frac{3}{2}|u_{2}|^{2}|v_{2}|^{2}+ \frac{3}{4}|v_{2}|^{4}\bigg{)}u_{2}+v_{2}+\lambda u_{1}+mv_{2 xx}=0,\] \[iv_{2t}-iv_{2x}+\bigg{(}\frac{1}{2}|v_{2}|^{2}+|u_{2}|^{2}\bigg{)} v_{2}-\eta\bigg{(}\frac{1}{4}|v_{2}|^{4}+\frac{3}{2}|v_{2}|^{2}|u_{2}|^{2}+ \frac{3}{4}|u_{2}|^{4}\bigg{)}v_{2}+u_{2}+\lambda v_{1}+mu_{2 xx}=0.\]  
In Equation (1), forward-propagating waves in core 1 and 2 are represented by \(u_{1,2}(x,t)\) and the backward propagating ones are denoted by \(v_{1,2}(x,t)\). \(m>0\) represents the strength of the dispersive reflectivity, \(\eta>0\) is the normalized quintic nonlinearity coefficient, and \(\lambda>0\) is the coefficient of the linear coupling between the cores. Based on the experimental values measured for the cubic-quintic nonlinearity for chalcogenide glasses [50; 51] and some organic materials [52; 53], it is found that \(\eta\) can range between 0.05 and 0.62. As a result, our analysis is confined to the range \(0\leq\eta\leq 1\). In addition, as was noted in Reference [40], \(m>0.5\) may not be physically significant. Therefore, the analysis will be limited to \(0\leq m\leq 0.5\).  
To determine the linear spectrum of the system, \(u_{1,2}\), \(v_{1,2}\)\(\sim\)\(\exp(i(kx-\omega t))\) are inserted into Equation (1). Linearizing the resulting equations and some straightforward algebraic manipulations lead to the following dispersion relation:  
\[\omega^{2}=\Big{(}1-mk^{2}\Big{)}^{2}+\lambda^{2}+k^{2}\pm 2\lambda\sqrt{ \left(1-mk^{2}\right)^{2}+k^{2}} \tag{2}\]The negative(positive) sign in (2) refers to the lower(upper) branch of the dispersion relation. When \(m=0\), the dispersion diagram relation reduces to the one in Reference [47]. The size of the bandgap can be determined from the roots of \(\frac{du^{2}}{dk^{2}}=0\). This leads to the following bandgap:  
\[\omega^{2}<\omega_{0}^{2}=\left\{\begin{array}{cc}(1-|\lambda|)^{2}&m\leq 0. 5,\\ &\\ \left(\frac{\sqrt{4m-1}}{2m}-|\lambda|\right)^{2}&m>0.5,\end{array}\right.\]  
where \(\omega_{0}^{2}\) is the minimum value of the \(\omega^{2}(k^{2})\). For \(m\leq 0.5\), \(\omega_{0}\) is attained at \(k_{0}^{2}=0\). On the other hand, for \(m>0.5\), \(\omega_{0}\) occurs at \(k_{0}^{2}=\frac{2m-1}{2m^{2}}\). It is worth noting that for \(0\leq m\leq 0.5\), the bandgap does not depend on \(m\) and it closes at \(\lambda=1\). Figure 1 shows the dispersion curves for different values of \(\lambda\) and \(m\).