Floquet solitons in square lattices: Existence, Stability and Dynamics

The paper establishes the phenomena numerically (existence of localized Floquet breathers; stability for θ=π near C≈1; a grid-size–independent complex quartet instability for θ=π/2 at C≈1.17; DNLS-like two-site stability classification), but it does not give rigorous proofs. The model’s solution aims to provide rigorous arguments; however, it contains substantial gaps: (i) an explicit far-field monodromy formula U_τ(C)=exp(−i(τ/2)CH) that contradicts the paper’s observation that the background monodromy is the identity at C=1 and predicts an arc half-angle α=2τ|C−1| instead of the paper’s α≈τ(C−1) near C=1; (ii) a reduction to a closed 4-site boundary-value problem that neglects resonant linear coupling to the lattice tail; and (iii) use of a Lyapunov–Perron construction in a unitary (non-hyperbolic) setting without non-resonance conditions. Thus, while the qualitative conclusions broadly agree with the paper’s numerics, the proposed proof strategy is incomplete and contains incorrect steps. Key paper claims remain numerically supported but unproven, and the model’s proof is not yet correct. Citations: background model and observations on α≈τ(C−1) and stability near C=1 for θ=π ; θ=π/2 quartet instability at C≈1.16–1.17 and its grid-size independence ; two-site in-/out-of-phase stability patterns ; overall claims and setup of the z-periodic coupling DNLS and switching functions Jk(z) .