Nonlinear spatial evolution of degenerate quartets of water waves

Conor Heffernan, Amin Chabchoub, Raphael Stuhlmeier

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper derives the same Hamiltonian system, boundary fixed-point phases, separatrix formulae η(Θ), tanh-laws for Θ(x), and the heteroclinic (maximum-depletion) condition β0 = −β1/2 with the logistic profile for η(x) as the candidate solution. Equations (28)–(33) present the Hamiltonian structure; (43)–(45) and (46)–(48) give the separatrices and tanh integration; and (49)–(50) give the vertical heteroclinic and logistic solution. There is a sign typo in the paper’s Eq. (40) for cos Θ2, but the usage elsewhere (e.g., Θ0 definition) is consistent with the candidate and the dynamics, confirming the model’s sign choice. See the cited passages in the PDF for the system, Hamiltonian, and explicit solutions ; the fixed-point equations including the sign typo appear in (39)–(41) .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript gives a rigorous, insightful Hamiltonian reduction of the spatial Zakharov equation in a three-mode truncation, classifies the phase portraits, and produces explicit breather/separatrix solutions including an optimal energy-conversion heteroclinic. The results are correct and useful for experiments and broader dispersive media. The only issues are a minor sign typo in one fixed-point expression and small opportunities to clarify existence conditions and denominators in separatrix formulae.