# Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quinic Nonlinearity
## 1 Introduction  
In the past few decades, significant attention has been given to fiber Bragg gratings (FBGs) owing to their applications in filtering, data format conversion, and dispersion compensation [1, 2, 3, 4, 5, 6, 7]. These applications have been realized, in part, due to a distinctive property of FBGs, namely the existence of a stopband (or photonic bandgap) in their linear spectrum [3, 8]. Additionally, FBGs possess a strong dispersion that can be a million times larger than the dispersion of silica [8, 9].  
In the case of FBGs written in nonlinear media (e.g., Kerr nonlinearity), the nonlinearity can be counterbalanced by the FBG's effective dispersion giving rise to the generation of gap solitons (GSs) [9, 10, 11]. Gap solitons have received much attention both theoretically and experimentally due to their potential applications in slow light applications, optical memory elements [12, 13], and optical logic operations [14, 15].  
Bragg grating solitons have extensively been studied in the cubic nonlinearity both theoretically [16, 17, 18, 19] and experimentally [20, 21, 22, 23]. It has been established that gap solitons in uniform Bragg gratings and Kerr nonlinearity exist throughout the bandgap and can be characterized by two parameters. Moreover, these gap solitons can travel at any velocity between zero (quiescent) and \(c/n\) where \(c\) is the speed of light in vacuum and \(n\) is the refractive index of the medium. As for stability of the gap solitons, it has been shown analytically and numerically that about 50% of the soliton family is stable [24, 25]. Experimentally, gap solitons with a speed of 23% of the speed of light have been observed [23].  
Dynamics of gap solitons have been considered in more complex nonlinear media and systems including photonic crystals [26, 27, 28], waveguide arrays [29], coupled systems [30, 31, 32], microcavities [33], quadratic nonlinear medium [34, 35], and cubic-quintic nonlinearity [36].  
The standard model of gap solitons assumes that the FBG is uniform and weak. However, for specialized gratings such as deep gratings, gratings based on photonic nanowires [37] and Bragg superstructures [38, 39], due to the inhomogeneity of the bandgapspectra, the effect of dispersion of reflectivity needs to be incorporated into the model [40]. Numerical simulations have demonstrated that dispersive reflectivity improves the stability of the moving and quiescent gap solitons and results in the enlargement of the stable region [40; 41].  
Soliton switching and multicomponent solitons in uniform optical media and structures such as nonlinear couplers are potential building blocks for optical signal processing devices [42; 43; 44; 45]. Indeed, the ability to perform optical switching and/or signal processing with gap solitons is an intriguing one. Such optical devices can be constructed by writing Bragg gratings in one or both cores of a fiber coupler [46]. As a result, gap solitons in grating-assisted couplers with identical and dissimilar Bragg gratings are of great interest [30; 31; 47]. It has been shown that generalization of these systems to include dispersive reflectivity results in a much richer dynamics which can be instrumental in designing novel optical devices [48].  
In Reference [47], the stability of GSs in coupled identical Bragg gratings with cubic-quintic nonlinearity was analyzed. In the present work, a generalization of the model of Reference [47] is considered which includes the effect of dispersive reflectivity in both gratings. We analyze the characteristics and dynamical stability of zero-velocity GSs in the generalized model. In Section 2, the generalized model and the features of the bandgap are discussed. In Section 3, the stationary quiescent soliton solutions and their characteristics are analyzed. The analysis of the stability of the quiescent GSs is presented in Section 4. A summary of the main results is provided in Section 5.