Floquet-engineered Topology-protected Nonlinear Gap Solitons in One-Dimensional Dimerized Lattices

We introduce and systematically characterize Floquet gap solitons, which are nonlinear localized wave modes supported by a one-dimensional dimerized lattice with periodically modulated nearest-neighbor couplings. Motivated by recent advances in topological photonics and ultracold atoms, we explore whether stationary gap solitons known from static discrete nonlinear Schrödinger (DNLS) systems can persist as stable, true Floquet-periodic solutions under strong periodic driving. Through careful numerical analysis, including self-consistent field (SCF) computations of static soliton profiles, split-step integration of the driven nonlinear dynamics, and rigorous eigenvalue stability assessments, we delineate the parameter space of coupling ratios and drive periods where these Floquet gap solitons robustly exist. Remarkably, our findings reveal that these solitons exhibit linear stability over wide parameter regions, remain highly localized even under moderate on-site disorder, and display quantized, one-site displacement during adiabatic cycling of the coupling ratio, thus extending Thouless pumping into the nonlinear domain. Our results confirm that key topological invariants persist in the presence of Kerr nonlinearity and dynamic modulation, offering new prospects for robust, topologically protected soliton transport and manipulation in experimental platforms such as photonic lattices and cold-atom arrays.