Existence of All Wilton Ripples of the Kawahara Equation

Ryan P. Creedon

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 Lyapunov–Schmidt reduction, auxiliary/bifurcation equations, and case-by-case outcomes (K=2, K=3, K≥4) are correct; the candidate solution independently reproduces the same reductions and leading-order balances (including the K=3 cubic and the K≥4 c̃r(0) formula) and offers an alternative, plausible high-order “cascade” justification for b(a) ≍ a^{K−3} instead of the paper’s reliance on external asymptotics. Overall, they align on statements and mechanisms, with a different final step in the K≥4 case.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript convincingly proves the existence and analyticity of all Wilton ripple branches for the Kawahara equation via a clean Lyapunov–Schmidt framework, a result of genuine scope in dispersive PDEs. The only external ingredient (high-order asymptotics for K≥4 to ensure nontriviality of the K-mode) is reasonable and well signposted, but a brief heuristic complement would improve readability. Computations and reductions are accurate, and the narrative is clear and accessible to specialists.