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On higher order isolas of unstable Stokes waves

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Pith's one-line read The paper computes sharp deep-water asymptotics for the coefficients $\beta_1^{(2)}$, $\beta_1^{(3)}$, $\beta_1^{(4)}$ that size the high-frequency instability isolas of Stokes waves, showing they decay exponentially and vanish only…

desk verdict Sharp deep-water asymptotics for the first three Stokes-wave isolas; p=2,3 are clean, p=4 hangs on a Mathematica cancellation that should be independently checkable. read the letter →

arxiv 2501.01390 v1 pith:FMCVGZIM submitted 2025-01-02 math.AP

Massimiliano Berti , Livia Corsi , Alberto Maspero , Paolo Ventura This is my paper · ORCID
classification math.AP MSC 35Q3535B3576B15
keywords Stokeswaveshigh-frequencyinstabilityisolasmodulationaldeepwaterlimitasymptoticexpansionspectraltheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Stokes waves are periodic traveling water waves, and a central question is whether small perturbations grow. The linearized problem has a sequence of closed spectral bands—isolas—away from the origin, indexed by $p\geq 2$, whose existence and size are controlled by an analytic function $\beta_1^{(p)}(h)$ of the water depth $h$. This paper proves that as $h\to+\infty$ the coefficients for $p=2,3,4$ decay exponentially, with the sharp leading terms $\beta_1^{(2)}\sim \frac{3\sqrt{3}}{64}e^{-h/2}$, $\beta_1^{(3)}\sim \frac{2\sqrt{2}}{3}e^{-2h}$, and $\beta_1^{(4)}\sim -\frac{5\sqrt{5}}{8\sqrt{3}}e^{-2h}$. Because the p-th isola has width proportional to $|\beta_1^{(p)}|\epsilon^p$, this quantifies how the first three high-frequency instability bands shrink in deep water. It also gives an independent proof that $\beta_1^{(p)}$ is not identically zero for $p=2,3,4$ and has only finitely many depth zeros, supporting a conjecture that the same is true for every $p$.

What carries the argument

The central object is the coefficient $\beta_1^{(p)}(h)$: the leading term in the discriminant $D^{(p)}(\mu,\epsilon)$ of the two eigenvalues of the Floquet operator $L_{\mu,\epsilon}$ near the double eigenvalue $i\omega_*^{(p)}(h)$ at $\mu=\varphi(p,h)$. When it is nonzero, the sign of $D^{(p)}$ changes across an interval of Floquet exponents of width proportional to $|\beta_1^{(p)}|\epsilon^p$, producing the isola. The paper computes its deep-water limit from explicit formulas in which $\beta_1^{(p)}$ is a sum of rational products of Stokes-wave Taylor-Fourier coefficients ($a_\ell^{[\ell]}$, $p_\ell^{[\ell]}$) and the quantities $\Omega_j^{(p)}$, $t_j^{(p)}$ built from the wavenumber $\varphi(p,h)$; Lemmas 2.3–2.5 refine the expansion of $\varphi(p,h)$ as $h\to\infty$, and the leading exponentials emerge after cancellations of the constant parts. For $p=4$ the cancellation of the eight three-intermediate-harmonic terms (equation (2.56)) is checked by a long computer-algebra computation.

What would settle it

Recompute the signed sum in (2.56) with independent symbolic or high-precision arithmetic: if the eight three-intermediate-harmonic terms do not cancel to $O(e^{-4h})$, the $p=4$ statement falls. Alternatively, evaluate $\beta_1^{(4)}(h)$ numerically at large $h$ (for example $h=20$) using the formulas of Section 2.3 and compare with $-\frac{5\sqrt{5}}{8\sqrt{3}}e^{-2h}$; a mismatch beyond the stated $O(e^{-4h})$ error would refute Theorem 2.1.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 2.1: in the deep-water limit $h\to+\infty$, the coefficient $\beta_1^{(p)}(h)$ that enters the discriminant of the two relevant eigenvalues of the Floquet operator has the asymptotic expansions $$\$beta_1^{{(2)}}$(h)=\frac{3\sqrt{3}}{64}$e^{{-h/2}}$+O($e^{{-3h/4}}$),\qquad \$beta_1^{{(3)}}$(h)=\frac{2\sqrt{2}}{3}$e^{{-2h}}$+O($e^{{-3h}}$),\qquad \$beta_1^{{(4)}}$(h)=-\frac{5\sqrt{5}}{8\sqrt{3}}$e^{{-2h}}$+O($e^{{-4h}}$).$$ Since the curve traced by the unstable eigenvalues is approximated by an ellipse of horizontal width proportional to $|\beta_1^{(p)}|\epsilon^p$, these formulas give the exact exponential rates at which the $p=2,3,4$ isolas shrink as the depth grows. The nonzero constants imply that each of these three coefficients is nonzero for all sufficiently large $h$, and, being real analytic, each has at most finitely many positive-depth zeros; the proof uses the explicit 27-term (for $p=4$) algebraic expressions for $\beta_1^{(p)}$ and refined asymptotics for the branching wavenumber.

Load-bearing premise

The $p=4$ expansion rests on a long computer-algebra cancellation, not written out in the paper, together with the correctness of the 27-term formula taken from the companion paper; if either is wrong, the stated leading constant $-\frac{5\sqrt{5}}{8\sqrt{3}}$ in (2.3) would change.

Editorial extensions

If this is right

  • For $p=2,3,4$, $\beta_1^{(p)}(h)$ is nonzero at all sufficiently large depths, so the corresponding high-frequency instability isolas exist for all deep enough water and do not accumulate an infinite sequence of critical depths.
  • The Floquet interval supporting the p-th isola has width $4|\beta_1^{(p)}|/T_1^{(p)}\,\epsilon^p + O(\epsilon^{p+1})$, hence in deep water it shrinks like $e^{-h/2}$ for $p=2$ and like $e^{-2h}$ for $p=3,4$.
  • The maximum growth rate of the unstable mode, of size $|\beta_1^{(p)}|\epsilon^p$, is exponentially small in the depth for these three bands, making the deep-water instability a weak but persistent effect.
  • Since $h\mapsto\beta_1^{(p)}(h)$ is real analytic, the expansions prove Conjecture 2.2 for $p=2,3,4$: each of these coefficients has only finitely many zeros in $(0,\infty)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same algebraic formulas could be expanded at finite depth to produce rigorous two-sided bounds on the isola widths, turning the numerical spectral plots into certified existence statements for each $p$.
  • The decay rates imply a concrete challenge for numerical solvers: for a tank of depth $h$, the $p=2,3,4$ eigenvalue gaps are of order $e^{-h/2}$ to $e^{-2h}$ times $\epsilon^p$, so direct detection requires resolving exponentially small separations.
  • The clean form of the $p=4$ constant, a rational multiple of $\sqrt{15}$, hints that the unshown cancellation in (2.56) may admit a human-readable derivation, which would make the $p=4$ case fully checkable without computer algebra.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the coefficient beta_1^{(p)}(h) that controls the width of the p-th high-frequency instability isola of Stokes waves in the deep-water limit. Building on the formula for beta_1^{(p)} as a sum of 3, 9, and 27 explicit terms for p=2,3,4 respectively, it proves Theorem 2.1, giving the sharp exponential decay rates beta_1^{(2)}~3*sqrt(3)/64*e^{-h/2}, beta_1^{(3)}~2*sqrt(2)/3*e^{-2h}, and beta_1^{(4)}~-5*sqrt(5)/(8*sqrt(3))*e^{-2h} as h->+infty. The p=2 and p=3 cases are derived analytically in the text, while the p=4 case relies on a Mathematica-assisted cancellation among the eight three-intermediate-harmonic terms. The paper also concludes that these beta_1^{(p)} have finitely many positive zeros, supporting Conjecture 2.2.

Significance. If the p=4 computation can be verified, the result provides sharp asymptotic widths for the p=2,3,4 isolas in deep water, confirms the exponential vanishing of beta_1^{(p)} at infinity, and establishes finiteness of zeros for these p. The p=2 and p=3 derivations are transparent and reproducible from the displayed formulas; the paper is also honest about the computational assistance used for p=4 and the analytic openness of p>4. The result is a meaningful complement to the existence theory in the authors' prior work [3] and lends support to a natural conjecture. The main caveat is that the p=4 branch rests on an unverified cancellation that is not documented in the text.

major comments (2)
  1. [Section 2.3, Eq. (2.56)] The assertion that the eight three-intermediate-harmonic terms sum to O(e^{-4h}) is load-bearing for the p=4 case of Theorem 2.1, but no derivation or intermediate output is given in the text; the only justification is 'a very long computation using Mathematica'. Since each individual term is of order one and has an e^{-2h} correction (via (2.41) and (2.44)), the cancellation of both the order-one and e^{-2h} parts is nontrivial. If the e^{-2h} coefficient of the signed sum were a nonzero c, formula (2.3) would become (-5*sqrt(5)/(8*sqrt(3))+c)e^{-2h}+O(e^{-4h}), changing the leading coefficient and the finiteness-of-zeros conclusion for p=4. Please provide the explicit expanded form of the eight terms to order e^{-4h}, or include a reproducible script with output, so that (2.56) can be checked without trusting an opaque computation.
  2. [Section 2.3, Eq. (2.48)] The two displayed expansions for [b(4)_1]^-_2 and [b(4)_1]^+_2 are printed identically, despite the different signs in their definitions (2.38c)-(2.38d). If this is a typo and the e^{-2h} coefficients differ, then (2.49) would not be O(e^{-4h}) and the leading coefficient in (2.3) would be affected. The authors should either verify that the two coefficients are indeed equal (explaining the cancellation) or correct the typo and rerun the computation.
minor comments (4)
  1. [Section 1, page 1] The name 'Zhakarov' should be 'Zakharov'.
  2. [References] Reference [12] (Creedon and Deconinck) is listed in the bibliography but never cited in the text.
  3. [Section 2.3, Lemma 2.5] The proof of Lemma 2.5 is only a reference to [3, formulas (A.11),(2.31)]; for self-containedness, a short derivation analogous to Lemmas 2.3-2.4 would help, since this expansion is a load-bearing input for the p=4 computation.
  4. [Section 2.3, Eq. (2.56)] If the identical expansions in (2.48) are intentional, a note explaining the cancellation would aid the reader; otherwise the identical appearance is a likely typo. More generally, a table listing the individual contributions of the 27 terms would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the exponential decay laws are new expansions of the fixed coefficient beta_1^(p) imported from the authors' prior work; the p=4 branch rests on an unshown Mathematica cancellation, a verification gap rather than circularity.

full rationale

The paper's central claim, Theorem 2.1, is an asymptotic expansion of the coefficient beta_1^(p)(h) for p=2,3,4. The starting point is the explicit formula for beta_1^(p) quoted from the authors' prior paper [3, (5.7)-(5.8)], together with the Stokes-wave Taylor coefficients taken from [5] and [6]. This is self-citation, but it is not circular: the exponential rates and leading constants in (2.1)-(2.3), such as 3*sqrt(3)/64, 2*sqrt(2)/3 and -5*sqrt(5)/(8*sqrt(3)), are not inputs to those formulas but are obtained by a new asymptotic computation from fixed analytic expressions. No fitted parameter is introduced, no predicted quantity is statistically forced by an earlier fit, and no uniqueness theorem or ansatz is imported from the authors' prior work to forbid alternatives. The only genuine weakness is in Section 2.3: equation (2.56) asserts that eight three-intermediate-harmonic terms sum to O(e^{-4h}) 'by a very long computation using Mathematica', without displaying the computation. If the e^{-2h} coefficient of that signed sum were nonzero, the leading coefficient in (2.3) would change. This is an omitted verification step and a correctness risk, but it is not a circularity: the asserted cancellation is independent of the result being derived, not equivalent to it. Therefore the paper has minor, non-load-bearing self-citation and no significant circular reasoning.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are introduced or fitted. The derivations use standard asymptotic expansions and quote explicit formulas for Stokes-wave Taylor-Fourier coefficients and for the wavenumbers from the authors' prior work [3,5,6]; the new content is a computation, not a new postulate.

assumptions (5)
  • domain assumption The formulas [3, (5.7)-(5.8)] give beta_1^{(p)} as a finite signed sum of the b-terms used in (2.4), (2.20), (2.37).
    Entered at the start of Sections 2.1-2.3; this is the entire input of the computation and is taken from the authors' prior paper [3].
  • domain assumption The Stokes-wave Taylor-Fourier coefficients a_ell^[ell] and p_ell^[ell] for ell=1,2,3,4 and their expansions (2.10), (2.24), (2.41) from [5], [6], [3] are correct.
    Used in all three cases; an error in these coefficients would alter the leading coefficient in Theorem 2.1.
  • domain assumption The wavenumber asymptotics phi(2,h), phi(3,h), phi(4,h) in Lemmas 2.3-2.5 hold to the stated accuracy; for p=4 the O(e^{-4h}) estimate is quoted from [3, Appendix A].
    Needed to expand Omega_j^{(p)} and t_j^{(p)}; a weaker remainder would not change the leading term but would weaken the error term in Theorem 2.1.
  • standard math tanh(M)=1-2e^{-2M}+O(e^{-4M}) for M to infinity and tanh(epsilon)=epsilon+O(epsilon^3) for epsilon to 0.
    Used throughout, first in equation (2.8), to expand hyperbolic tangent factors at large depth.
  • standard math A non-vanishing analytic function on (0, infinity) with nonzero limits at both ends has finitely many zeros.
    Used after Theorem 2.1 to conclude that beta_1^{(p)} has finitely many zeros for p=2,3,4.

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Pith. "Pith review of On higher order isolas of unstable Stokes waves." pith.science (2026). https://pith.science/paper/FMCVGZIM

@misc{pith2026250101390,
  author       = {Pith},
  title        = {Pith review of: On higher order isolas of unstable Stokes waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMCVGZIM}},
  note         = {Machine review of arXiv:2501.01390}
}
abstract

We overview the recent result [3, Theorem 1.1] about the high-frequency instability of Stokes waves subject to longitudinal perturbations. The spectral bands of unstable eigenvalues away from the origin form a sequence of {\it isolas} parameterized by an integer $ \mathtt{p} \geq 2 $ for any value of the depth $ \mathtt{h} > 0 $ such that an explicit analytic function $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not zero. In [3] it is proved that the map $ \mathtt{h} \mapsto \beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not identically zero for any $ \mathtt{p} \geq 2 $ by showing that $ \lim_{\mathtt{h} \to 0^+}\beta_1^{(\mathtt{p})}(\mathtt{h}) = - \infty $. In this manuscript we compute the asymptotic expansion of $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ in the deep-water limit $ \mathtt{h} \to + \infty $ -- it vanishes exponentially fast to zero -- for $\mathtt{p}=2$, $3$, $4$.

Figures

Figures reproduced from arXiv: 2501.01390 by the authors.

Figure 1
Figure 1. Spectral bands with non zero real part of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The instability region around the curve µ (p) 0 (ϵ) delimited by the curves µ (p) ∧ (ϵ) and µ (p) ∨ (ϵ). 1. Upper bounds: In view of (1.11) and (1.9), for any p ≥ 2 and any depth h > 0, the real part of the eigenvalues (1.10) is at most of size O(ϵ p ) implying that the isolas, if ever exist, shrink exponentially fast as p → +∞. 2. Lower bounds: In view of (1.10), (1.11), for any p ≥ 2 and any depth h > 0, a suffici… view at source ↗
Figure 3
Figure 3. Case p = 2. Plot of the function β (2) 1 (h) (in blue), vanishing at the red point h∗ ≈ 1.84940. In yellow the leading part of its asymptotic expansion in the deep-water limit [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Case p = 3. Plot of the function β (3) 1 (h) (in blue), vanishing at the red point h∗ ≈ 0.82064. In yellow the leading part of its asymptotic expansion in the deep-water limit [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Case p = 4. On the left: plot of the function β (4) 1 (h) (in blue) vanishing at the red points h∗ ≈ 0.566633 and h∗∗ ≈ 1.255969. On the right: zoom on the asymptotic behavior of β (4) 1 (h) in blue, in yellow the leading part of its expansion in the deep-water limit. …

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