Fronts in Dissipative Fermi-Pasta-Ulam-Tsingou Chains

The paper proves existence, uniqueness (with phase), monotonicity, and sharp exponential tails for the nonlocal front equation via an implicit function theorem in weighted Sobolev spaces and a careful Fourier-symbol analysis; see Theorem 1 and Sections 4–5, including the O(ε) estimate in H1 and the tail analysis through the modified kernels a±ε and their poles . The candidate solution correctly matches several ingredients (limit ODE, small-ε continuation, O(ε) estimate, symmetry), but its monotonicity and tail arguments hinge on an unstated “maximum principle/barrier” for the operator Λε∗ρ′ + ρ − Φ′′(0)Λε∗ρ, which is not positivity-preserving due to the odd kernel entering via Λε∗ρ′. The paper avoids this pitfall by re-casting the problem for Sε = −Rε′ and exploiting resolvent kernels a±ε; it then proves positivity and precise rates without invoking a maximum principle . Because the model’s monotonicity step relies on an invalid comparison principle for the nonlocal operator, its proof is incomplete at a crucial point, and thus incorrect as written.