Floquet solitons in square lattices: Existence, stability, and dynamics

Abstract

In the present work, we revisit a recently proposed and experimentally realized topological two-dimensional lattice with periodically time-dependent interactions. We identify the fundamental solitons, previously observed in experiments and direct numerical simulations, as exact, exponentially localized, periodic in time solutions. This is done for a variety of phase-shift angles of the central nodes upon an oscillation period of the coupling strength. Subsequently, we perform a systematic Floquet stability analysis of the relevant structures. We analyze both their point and their continuous spectrum and find that the solutions are generically stable, aside from the possible emergence of complex quartets due to the collision of bands of continuous spectrum. The relevant instabilities become weaker as the lattice size gets larger. Finally, we also consider multisoliton analogs of these Floquet states, inspired by the corresponding discrete nonlinear Schrödinger (DNLS) lattice. When exciting initially multiple sites in phase, we find that the solutions reflect the instability of their DNLS multi-soliton counterparts, while for configurations with multiple excited sites in alternating phases, the Floquet states are spectrally stable, again analogously to their DNLS counterparts.

Period-doubled Floquet Solitons

“…In such media, nonlinear effects arise naturally at high optical intensities via the optical nonlinearity available in the ambient mediumfor which the Kerr effect is the relevant one in this experiment. Some recent examples of nonlinear topological effects include the formation of Floquet solitons in a topological bandgap [26][27][28], topological edge solitons [29][30][31], nonlinearity-induced local topological edge states [32], and nonlinear Thouless pumping [33,34]. In this context, we also highlight recent works on interacting Floquet polaritons [35], modulation-induced nonlinearity in magnetic insulators [36] and optical devices [37].…”

“…Spatial solitons are localized shape-preserving waves formed when nonlinearity balances linear diffraction of a wave packet [38][39][40][41]. Floquet solitons are nonlinear waves in the Floquet sense: Due to the periodic nature of the system's Hamiltonian, they reproduce themselves with each driving period, up to a phase factor [26][27][28].…”

Standing and traveling waves in a model of periodically modulated one-dimensional waveguide arrays

“…From here on, we assume both initial conditions are nonzero. The analysis that follows is an adaptation of the method used in [30,31]. We define…”