The existence, stability, and collisions of moving solitons in Bragg gratings (BGs) in a cubic-quintic nonlinear medium are investigated. Two disjoint families of solitons that are separated by a border are identified. One family (Type 1) can be considered as the generalization of the moving solitons in BGs written in a cubic nonlinear medium. The other family (Type 2) occurs in regions where quintic nonlinearity dominates. Through systematic numerical stability analysis, the stability regions in the plane of quintic nonlinearity versus frequency have been determined. It is found that the stability regions are dependent on the velocity of solitons. The collisions of counterpropagating solitons have been systematically investigated. Collisions of in-phase Type 1 solitons can result in a variety of outcomes including forming two asymmetrically separating solitons and passing through each other and separating symmetrically with reduced, unchanged, or increased velocities. In certain parameter regions, solitons merge to form a quiescent one. An outcome that has not been reported previously for uniform gratings is the formation of a quiescent soliton and two symmetrically separating solitons. This outcome is found to be more robust than the merger.
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