Steady waves in flows over periodic bottoms

Carlos Garc\'ia-Azpeitia; Walter Craig

REVIEW 2 major objections 4 minor 27 references

Steady waves in flows over periodic bottoms

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a non-degenerate flat-bottom wave, a small periodic bottom yields at least two phase-shifted steady waves.

desk verdict The two-wave persistence theorem is a real conditional result; the advertised Stokes-wave consequence and the Theorem 12 parameter estimate both overreach. read the letter →

arxiv 1908.03787 v7 pith:SSAQNSED submitted 2019-08-10 math.AP

Walter Craig , Carlos García-Azpeitia This is my paper · ORCID

classification math.AP MSC 76B1535B3235Q35

keywords waterwavessteadyperiodicbottomDirichlet-NeumannoperatorStokessymmetrybreakingLyapunov-SchmidtreductionHamiltonianformulation

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

Steady water waves over a wavy bottom are obtained as perturbations of flat-bottom waves, and the perturbation does not destroy the waves: it multiplies them. The paper proves that for any non-degenerate flat-bottom steady wave $u_c$—one whose only neutral direction is the horizontal translation—a sufficiently small periodic bottom $b$ admits at least two steady waves $u_{b,1}=u_c(x+\theta_1)+O(\|b\|)$ and $u_{b,2}=u_c(x+\theta_2)+O(\|b\|)$ with distinct phases $\theta_1,\theta_2$. This applies, conditionally on a non-degeneracy assumption, to Stokes waves on the primary bifurcation branch, giving the first rigorous multiplicity statement of this kind. The same Hamiltonian machinery also recovers the unique continuation of the trivial solution for small bottoms except at the critical speeds $c_k$. The underlying reason is topological: a circle-valued reduced energy must have a maximum and a minimum, so symmetry breaking forces at least two steady configurations rather than none.

What carries the argument

The machinery has three parts. First, the Dirichlet-Neumann operator $G(\eta;b)$ turns the free-boundary Euler equations into an analytic Hamiltonian $H(\eta,\xi;b,c)$ in surface-elevation and velocity-trace coordinates, with a uniform current term of mean speed $c$. Second, for flat bottoms the Hamiltonian is invariant under horizontal translations $S^1$, so a non-constant solution $u_c$ sits on a circle of translated copies; the slice coordinate $\upsilon(\theta,w)=\theta\cdot(u_c+w)$ ($\theta\in S^1$, $w$ orthogonal to $\partial_x u_c$) provides local coordinates near that circle. Third, Lyapunov-Schmidt reduction solves the normal equation $\nabla_w H_b(\theta,w)=0$ by the implicit function theorem, yielding the reduced one-dimensional Hamiltonian $h_b(\theta)=H_b(\theta,w(\theta;b))$ on $S^1$. The key mechanism is that $h_b$ is a small non-constant perturbation of a constant function, and any such function on a circle has at least two critical points—its maximum and minimum—giving the two steady waves.

What would settle it

If along the finite-depth Stokes branch one finds a wave where the Hessian $D^2H(u_c;0,c)$ has a zero mode on the orthogonal complement of $\partial_x u_c$, the non-degeneracy hypothesis fails at that wave and Theorem 16 gives no two-wave conclusion there.

Watch

Extended reading notes

Core claim

The central claim is Theorem 16. Start with a nontrivial steady wave $u_c$ for a flat bottom whose set of horizontal translates—its $S^1$-orbit—is non-degenerate: at $u_c$ the Hessian of the Hamiltonian is invertible on the subspace of functions orthogonal to $\partial_x u_c$. The theorem states that under this hypothesis every sufficiently small periodic bottom perturbation $b$ produces at least two steady solutions $u_{b,j}(x)=u_c(x+\theta_j)+O(\|b\|_{s+1})$ with distinct phases $\theta_j$. In the proof, the Hamiltonian is written in slice coordinates $\upsilon(\theta,w)=\theta\cdot(u_c+w)$ near the orbit; the implicit function theorem eliminates the normal component $w$, and the reduced Hamiltonian $h_b(\theta)$ on the circle $S^1$ is a small deformation of a constant. A continuous function on a circle must have a maximum and a minimum, each giving one of the two waves. Applied to the classical Stokes waves—periodic traveling gravity waves of permanent form—this yields persistence of two steady waves over periodic bottoms along the primary branch, away from degenerate points.

Load-bearing premise

The result depends on the flat-bottom wave being isolated up to horizontal translations: no small deformation orthogonal to the translation direction may solve the linearized problem. For finite-depth Stokes waves this is assumed from infinite-depth results and numerics, and the paper says the finite-depth proof 'may be possible' rather than supplying it.

Editorial extensions

If this is right

For any non-degenerate flat-bottom wave $u_c$, each sufficiently small periodic bottom $b$ supports at least two distinct steady waves, both of the form $u_c(x+\theta_j)+O(\|b\|_{s+1})$.
Away from the critical speeds $c_k$, the flat trivial solution has a unique continuation to small bottoms on the explicit parameter set $|c-c_k|\ge k^{-3/2}\varepsilon$, $|b|_{s+1}\le\varepsilon^{2(1+\delta)}$; this quantifies the previous continuation theorem.
When $u_c$ and $b$ share the period $2\pi/p$, the two persisting waves come in $\mathbb{Z}_p$-orbits of phase shifts, so counting spatially distinct profiles multiplies the two solutions by $p$.
Assuming the infinite-depth non-degeneracy proof extends to finite depth via a Babenko-type formulation, the primary Stokes branch supports two steady waves over a wavy bottom at every non-degenerate point, with only a discrete exceptional set.

Reading between the lines

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

One testable prediction is the phase relation: for a small sinusoidal bottom, the maximum and minimum of $h_b$ should sit near the bottom's crest and trough, so the two waves are approximately in phase and anti-phase with the bed—the dune and antidune patterns mentioned in the introduction.
Since any nonconstant function on a circle has at least two critical points, the 'two waves' count is the generic minimum; bottoms with several Fourier modes could produce additional extrema of $h_b$ and hence more than two steady waves near the same orbit.
A close numerical study of the restricted Hessian along the finite-depth Stokes branch as amplitude grows would identify the exact points where non-degeneracy, and therefore the two-wave conclusion, can fail; this is the sharpest route to making the theorem unconditional.
For higher-dimensional tori $\mathbb{T}^n$ with surface tension, the same reduction yields a reduced Hamiltonian on $\mathbb{T}^n$, and Lusternik-Schnirelmann category should force at least $n+1$ critical points—so the mechanism generalizes from 'two waves' to 'several waves' as the dimension of the symmetry group grows.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Let me give you the quick read on Craig & García-Azpeitia, arXiv:1908.03787. Two things to know before you open it. First, the main result—Theorem 16—is a legitimate new conditional theorem: from a non-degenerate S^1 orbit of a flat-bottom steady wave, a small periodic bottom perturbation yields at least two nearby steady waves at distinct phases. The proof is a standard Lyapunov-Schmidt reduction plus the observation that a continuous function on a circle has a max and a min. That part is sound, and it is genuinely new as far as I can tell. Second, the paper overreaches in two places. The abstract says the theorem gives persistence of two Stokes waves; the actual hypothesis needed—non-degeneracy of the orbit—is not proven for finite-depth Stokes waves. They cite the infinite-depth result of Buffoni–Dancer–Toland and numerical evidence, and only say a finite-depth extension 'may be possible.' So the 'Consequently...' line is stronger than what is proved. The other soft spot is Theorem 12. The proof claims a uniform lower bound of order k for the eigenvalues λ^-_k, but their own derivative computation gives |∂_c λ^-_k| ≈ √k. Since their parameter region excludes only a neighborhood of width k^{-3/2}ε around each c_k, the linear estimate yields a lower bound of order ε/k, not kε. That does not give the uniform invertibility bound they need. This is a real error, and it undermines the continuation from the trivial solution and Corollary 13. It does not, however, touch Theorem 16, which is independent of the uniform-in-k bound.

Credit where due: the paper is honest about the non-degeneracy caveat in the introduction, and the slice construction for the infinite-dimensional action is handled cleanly. The main theorem is likely correct and is a useful addition to the water-waves toolbox. The citation pattern looks fine; the prior work is acknowledged.

Bottom line: send it to a serious referee. It needs revision—fix the eigenvalue estimate or weaken the claim, and adjust the abstract to say 'if the non-degeneracy holds.' But the central persistence result is worth printing.

Referee Report

2 major / 4 minor

Summary. The paper studies steady two-dimensional water waves in a periodic strip with a periodic bottom perturbation b and a mean current c. Using a Hamiltonian formulation based on the Dirichlet-Neumann operator, it proves (Theorem 9) unique continuation of the trivial solution for small b when c is not one of the linear speeds c_k, and (Theorem 12) a quantitative version of this continuation under a smallness condition |b| ≤ ε^{2(1+δ)} outside c-neighborhoods of size k^{-3/2}ε. The main result (Theorem 16) states that if a flat-bottom steady wave u_c has a non-degenerate S^1-orbit in the sense of Definition 15, then for small periodic b there are at least two nearby steady waves u_{b,1}=u_c(x+θ_1)+O(|b|) and u_{b,2}=u_c(x+θ_2)+O(|b|). The proof uses slice coordinates and a Lyapunov-Schmidt reduction to a Hamiltonian on S^1, whose maximum and minimum give the two phases.

Significance. The main idea is elegant: breaking translation symmetry by a small periodic bottom turns the S^1-orbit into a reduced Hamiltonian on a circle, so the existence of at least two critical points follows from elementary topology. Theorem 16 is proved by a clean, internally consistent Lyapunov-Schmidt reduction, and the non-degeneracy hypothesis is explicitly stated. The appendix also gives a useful self-contained construction of the harmonic function Φ_b and estimates for the Dirichlet-Neumann operator. The value of the paper, however, is conditional: the advertised Stokes-wave application rests on a non-degeneracy assumption that is not proven for finite-depth Stokes waves, and the quantitative continuation theorem contains a spectral estimate that is incorrect as written. These issues do not destroy the main theorem, but they require revision.

major comments (2)

[§3.2, Theorem 12] The spectral estimate in the proof of Theorem 12 is incorrect as written. From the displayed formula for λ^-_k one obtains ∂_c λ^-_k(c_k) = -2 c_k k^2 / sqrt((g-k tanh(hk))^2 + 4 c_k^2 k^2), so ∂_c λ^-_k(c_k)/√k → -2√g; the expression in the paper has the factor √k in the numerator rather than the denominator and the denominator lacks the square root. Consequently, the claimed lower bound |λ^-_k(c)| ≥ k γ1 ε for |c-c_k| ≥ k^{-3/2}ε is not correct in order of magnitude: the mean value theorem gives |λ^-_k(c)| ∼ ε/k in the worst case. Since the velocities c_k accumulate at c=0, the uniform invertibility estimate ‖L(c)^{-1}‖_{Y→X} ≤ 1/(γ1 ε) is not established; in H^s norms the inverse can grow like k^2/ε for large k. Theorem 12 and Corollary 13 therefore need revision, for example by keeping c uniformly away from 0 or by letting the admissible ε depend on the mode k.
[Abstract and §1 (Main Result)] The abstract states, as a consequence, that the paper obtains persistence of at least two steady waves close to a non-degenerate S^1-orbit of Stokes waves bifurcating from the velocities c_k. However, non-degeneracy in the sense of Definition 15 is not established for finite-depth Stokes waves. The introduction itself says only that it 'may be possible' to extend the infinite-depth result [2] using the Babenko-like formulation [20], and the numerical evidence [8] is not a proof. Thus the Stokes-wave conclusion is conditional on an open hypothesis. The abstract and the 'Main Result' paragraph should be rephrased so that this hypothesis is explicit, rather than presenting the consequence as unconditional.

minor comments (4)

[Abstract] The word 'non-trival' should be 'non-trivial'.
[Figure 2 and §1] The symbols c_* and λ_j are used in the introduction and figure captions without precise definitions; they should be defined or the text should refer to the corresponding hypothesis.
[Theorem 16] The statement writes the perturbation as O(|b|_{s+1}) without specifying the space; the proof makes clear it is O(|b|_{s+1}) in X, and the statement should say this explicitly.
[§3.2, proof of Theorem 12] The line 'Since the theorem holds uniformly in the region of parameters ... for any δ>0' should be reworded after the spectral estimate is corrected, because the uniformity in k is precisely the point that needs justification.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 16 is a self-contained conditional reduction; the Stokes-wave application is conditional on an unproven hypothesis, not a fitted or predicted quantity.

full rationale

I find no circular step in the derivation chain. Theorem 16 is a self-contained conditional result: given Definition 15 (invertibility of the restricted Hessian D^2_w H(u_c;0,c) on W), the proof constructs Palais-slice coordinates, solves the normal equation by the implicit function theorem, and forms a reduced Hamiltonian h_b:S^1->R. Critical points of h_b correspond to critical points of the full Hamiltonian because w solves the normal equation, and the maximum and minimum of a continuous function on S^1 supply at least two distinct critical points. The 'at least two' conclusion is not contained in the non-degeneracy assumption; it comes from the topology of S^1 plus the reduction. No parameter is fitted to data and then renamed a prediction. The Hamiltonian formulation from [7] and the analyticity of the Dirichlet-Neumann operator from [21] are external, parameter-free inputs with stated assumptions that do not include the target result, so their use is not circular. The only caveat is that the advertised application to the Stokes branch requires non-degeneracy of the primary Stokes wave, which the paper itself flags: 'It may be possible to extend the results in [2] for a flat bottom using the Babenko-like formulation in [20], which would imply that the primary branch of Stokes waves has non-degenerate S1-orbits except for a discrete set of values.' That makes the application conditional, not circular. Any eigenvalue-bound defect in Theorem 12 would affect Corollary 13 rather than Theorem 16 and is a correctness matter, not a circularity matter.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central theorem is conditional on the non-degeneracy hypothesis, which is the only significant unproved input. No free parameters are fitted and no new physical entities are introduced. The Hamiltonian and Dirichlet-Neumann operator framework is carried over from prior work.

assumptions (4)

domain assumption Existence of a non-degenerate S^1-orbit of a non-trivial steady wave u_c for the flat bottom (Definition 15)
Hypothesis of Theorem 16; not proven in this paper for finite-depth Stokes waves, only cited from [2] (infinite depth) and [8] (numerical).
standard math Analyticity and boundedness of the Dirichlet-Neumann operator G(eta;b) for bounded |eta| and |b| (as in Lannes, Theorem A.11)
Imported from [21]; used throughout the Hamiltonian formulation and in estimates of Section 3.
domain assumption Critical points of the Hamiltonian H(eta,xi;b,c) correspond to steady solutions of the Euler equation for b != 0 (Remark 6)
The paper notes that (eta,xi) are not canonical for b != 0, but grad H = 0 is still equivalent to steady Euler solutions; this is a physical modeling fact.
standard math The local diffeomorphism property of the special Palais-slice coordinates upsilon(theta,w) = theta*(u_c+w) on X = H^{s+1}_0 x H^{s+1}_0
Based on the implicit function theorem and G-equivariance; the paper argues this is an easy consequence of isometric action.

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Cite this review

Pith. "Pith review of Steady waves in flows over periodic bottoms." pith.science (2026). https://pith.science/paper/SSAQNSED

@misc{pith2026190803787,
  author       = {Pith},
  title        = {Pith review of: Steady waves in flows over periodic bottoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SSAQNSED}},
  note         = {Machine review of arXiv:1908.03787}
}

abstract

We study the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution when a flat bottom is perturbed, except for a sequence of velocities $c_{k}$. The main contribution is the proof that at least two steady solutions exist close to a non-degenerate $S^{1}$-orbit of non-constant steady waves when a flat bottom is perturbed. Consequently, we obtain persistence of at least two steady waves close to a non-degenerate $S^{1}$-orbit of Stokes waves bifurcating from the velocities $c_{k}$.

Figures

Figures reproduced from arXiv: 1908.03787 by the authors.

**Figure 2.** Left: Bifurcation diagram of the family of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Buﬀoni, E

B. Buﬀoni, E. Dancer, J. Toland. The sub-harmonic bifurcation of Stokes waves. Arch. Rational Mech. Anal. 152 (2000) 241–271

work page 2000

[2] [20]

Kuznetsov, E

N. Kuznetsov, E. Dinvay. Babenko’s Equation for Periodic Gravity Waves on Water of Finite Depth: Derivation and Numerical Solution. Water Waves 1(2019) 41–70

work page 2019

[3] [8]

Craig , D

W. Craig , D. Nicholls. Traveling gravity water waves in two and three dimensions. European Journal of Mechanics B/Fluids 21 (2002) 615– 641

work page 2002

[4] [1]

Amick, L

C. Amick, L. Fraenkel, J. Toland. On the Stokes conjecture for the wave of extreme form . Acta Math. 148 (1982) 193–214

work page 1982

[5] [3]

Brooke Benjamin, P

T. Brooke Benjamin, P. Olver. Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125 (1982) 137-185

work page 1982

[6] [4]

W. Craig. On the Hamiltonian for water waves. (arXiv:1612.08971)

work page Pith review arXiv

[7] [5]

Craig, M

W. Craig, M. Gazeau, C. Lacave, C. Sulem. Bloch Theory and Spectral Gaps for Linearized Water Waves. SIAM J. Math. Anal., 50(5) (2018) 5477–5501

work page 2018

[8] [6]

Craig, P

W. Craig, P. Guyenne, D. Nicholls, C. Sulem. Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005) 839–873

work page 2005