Localized patterns in planar bistable weakly coupled lattice systems

The paper’s Theorem 1 establishes, in the anti-continuum regime, (i) persistence of on-site and off-site D4-symmetric steady states in each component of U_{δ*}(Γ(N*)) \ U_{2δ*}(E) with uniqueness, smooth parameter dependence, and limits as d→0 lying in Γ(N*); (ii) nonlinear stability for branches continuing ū(N,M)(μ) on μ∈(δ*,1−δ*) and instability for branches continuing v̄(N,M)(μ) with an explicit unstable-eigenvalue count; and (iii) exactly four (when M=N−1) or eight (when M≠N−1) eigenvalues cross transversely at folds near μ=0 and μ=1, with no other imaginary spectrum. These are stated and proved via Implicit Function Theorem and Lyapunov–Schmidt reductions, together with careful near-fold expansions (Hypothesis 1; Theorem 1; Lemma 3.1; Lemmas 3.2–3.5) . The candidate solution proves the same three items, but with a different route: a Banach-space IFT on ℓ∞ for persistence; Gershgorin discs and linearized-stability principle for nonlinear stability; and a finite-dimensional Schur-complement/Lyapunov–Schmidt count for unstable and crossing eigenvalues, with the 8-versus-4 D4-orbit count. These steps match the paper’s conclusions (e.g., the 8 vs 4 eigenvalue counts near μ=1 and μ=0 and transverse crossings) . Minor methodological differences include the model’s explicit use of Gershgorin and Schur complements versus the paper’s scaled reductions. No substantive contradictions were found; both arguments are logically correct on the region that excludes E, and rely on the same Hypothesis 1 setup and D4-symmetry framework (index set I) .