# Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quinic Nonlinearity
## 3 Quiescent Gap Soliton Solutions  
For \(m=0\), Equation (1) simplify to the model of Reference [47] which admits exact analytical soliton solutions. However, for \(m\
eq 0\), there is no analytical solution to Equation (1) and the soliton solutions have to be obtained via numerical methods. To this end, we assume that the quiescent soliton solutions can be written as \(u_{1,2}(x,t)=U_{1,2}(x)e^{-i\omega t}\) and \(v_{1,2}(x,t)=V_{1,2}(x)e^{-i\omega t}\). Substituting these expressions into Equation (1) and simplifying leads to the following system of differential equations:  
\[-mU_{1 xx}=\omega V_{1}-iV_{1x}+\left(\frac{1}{2}|V_{1}|^{2}+|U_{1}|^{2}\right) V_{1}-\eta\left(\frac{1}{4}|V_{1}|^{4}+\frac{3}{2}|V_{1}|^{2}|U_{1}|^{2}+ \frac{3}{4}|U_{1}|^{4}\right)V_{1}+U_{1}+\lambda V_{2},\]  
\[-mV_{1 xx}=\omega U_{1}+iU_{1x}+\left(\frac{1}{2}|U_{1}|^{2}+|V_{1}|^{2}\right) U_{1}-\eta\left(\frac{1}{4}|U_{1}|^{4}+\frac{3}{2}|U_{1}|^{2}|V_{1}|^{2}+ \frac{3}{4}|V_{1}|^{4}\right)U_{1}+V_{1}+\lambda U_{2},\]  
\[-mU_{2 xx}=\omega V_{2}-iV_{2x}+\left(\frac{1}{2}|V_{2}|^{2}+|U_{2}|^{2}\right) V_{2}-\eta\left(\frac{1}{4}|V_{2}|^{4}+\frac{3}{2}|V_{2}|^{2}|U_{2}|^{2}+ \frac{3}{4}|U_{2}|^{4}\right)V_{2}+U_{2}+\lambda V_{1},\]  
\[-mV_{2 xx}=\omega U_{2}+iU_{2x}+\left(\frac{1}{2}|U_{2}|^{2}+|V_{2}|^{2}\right) U_{2}-\eta\left(\frac{1}{4}|U_{2}|^{4}+\frac{3}{2}|U_{2}|^{2}|V_{2}|^{2}+ \frac{3}{4}|V_{2}|^{4}\right)U_{2}+V_{2}+\lambda U_{1}.\]  
The relaxation algorithm is used to solve Equation (3). The relaxation algorithm is a numerical technique for solving boundary value problems. The algorithm employed in our analysis utilizes a deferred correction technique together with Newton iteration (for further details on the relaxation algorithm, see References [54; 55]). Similar to the model of Reference [47], two disjointed families of zero-velocity solitons are found in the model of Equation (1), namely, Type 1 and Type 2 solitons. Furthermore, both asymmetric \((U_{1}\
eq U_{2},V_{1}\
eq V_{2})\) and symmetric \((U_{1}=U_{2},V_{1}=V_{2})\) solitons exist in these soliton families. Type 1 and Type 2 solitons differ in their phase structure and the amplitudes of the solitons. More specifically, for both symmetric and asymmetric Type 1 solitons, the real(imaginary) parts of the forward and backward waves are even(odd) functions of \(x\). However, for symmetric and asymmetric Type 2 solitons, the real and imaginary parts of the the forward and backward waves have opposite parities compared with those of the Type 1 ones. The boundary that separates the soliton families has to be found numerically. It is found that the boundary is approximated very well by the expression \(1-\omega-\lambda=\dfrac{27}{160\eta}\). It should be noted that no soliton solutions exist on the boundary. Typical profiles for Type 1 and Type 2 symmetric solitons are displayed in Figure 2. In the single core and dual-core gap soliton models without dispersive reflectivity (see References [36; 47]), the Type 2 solitons have a sharp (but nonsingular) peak and close to the boundary between the soliton families, they possess a two-tier structure. However, our analysis shows that the profiles of Type 2 asymmetric and symmetric solitons in the present model are drastically different in that their peaks are not sharp and the two-tier structure is absent. Another observation is that, similar to the other models with dispersive reflectivity, there exists a critical value of \(m\) beyond which the solitons develop sidebands.