Multi-front dynamics in spatially inhomogeneous Allen-Cahn equations

Robbin Bastiaansen, Arjen Doelman, Tasso J. Kaper

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorem 5.3, its Melnikov asymptotics (Lemma 5.1), and the interaction ODE (5.2) establish the one-front count and stability, the N-front enumerations for localized topographies with exponentially decaying tails, and instability of all stationary N-fronts for N ≥ 2. The candidate solution reaches the same conclusions and uses essentially the same ingredients (Melnikov function S, large-gap scaling, coding of choices). However, it inserts extra “endpoint ghost-force” terms into the telescoping difference form for the stationary reduction that do not appear at leading order in the paper’s ODE (5.2); those tail effects are already captured inside S(ψ1) and S(ψN). Despite this derivation flaw, the candidate’s final statements (counts, μ*(N), stability) match the paper’s results.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript synthesizes Melnikov theory and front-interaction ODEs to deliver explicit counts and instability results for stationary multi-fronts under localized and periodic topographies. While the derivations are standard at leading order, the presentation is clear and the results are sharply formulated. Some technical steps, particularly the role of higher-order corrections behind the explicit threshold μ*(N), could be explained more fully to bolster rigor and reader confidence.