2407.04320
Asymptotic Analysis of a bi-monomeric nonlinear Becker-Döring system
Marie Doumic, Klemens Fellner, Mathieu Mezache, Juan J.L. Velázquez
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Proposition 4.3 states the two-phase asymptotics Ln ∼ sqrt(2An/ε), En ∼ 1 for 1 ≪ n ≪ 1/ε, and Ln ∼ (1/ε)(1 − e^{-1}e^{-Aεn}), En ∼ e^{-1}e^{-Aεn} for 1/ε ≪ n ≲ −(log ε)/ε, with A defined by (67) and derived from the cycle map (50) refined to (70) and then a continuous-in-s ODE system (72)–(75) that yields the invariant e+ℓ=1 and the e^{-1} amplitude. The candidate solution derives the identical ODE reduction from the same cycle-to-cycle laws, recovers the invariant E+εL=const, integrates an equivalent implicit relation, and obtains the same two regimes and amplitudes. The positivity of A invoked by the model follows from the paper’s boundary-layer formulation and matching (56)–(58), (64)–(66), where G, U and M are nonnegative. Overall, the model reproduces the paper’s argument closely and reaches the same asymptotics.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The asymptotic analysis for Phases I–II is consistent and well organized. The reduction from the discrete cycle map to an ODE system, the invariant, and the implicit solution are clearly presented and align with standard methods. However, parts of the argument remain heuristic, and the key boundary-layer ingredients (existence/uniqueness/positivity of U and M) are deferred to a companion paper. Adding brief clarifications and conditions would improve rigor and readability without altering the main conclusions.