pith. sign in

arxiv: 2604.12269 · v1 · submitted 2026-04-14 · ⚛️ physics.flu-dyn · math-ph· math.MP

Recurrent bifurcations of stability spectra for steep Stokes waves in a deep fluid

Sergey Dyachenko , Robert Marangell , Dmitry E. Pelinovsky This is my paper

Pith reviewed 2026-05-10 15:47 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math-phmath.MP
keywords Stokes wavesmodulational stabilitybifurcationsdeep fluidpseudo-differential operatorsconformal variablesFloquet parameternormal forms
0
0 comments X

The pith

Stokes waves exhibit four recurrent bifurcations in their stability spectra as steepness increases to the limiting peaked wave.

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

The paper derives criteria and normal forms for four types of bifurcations in the modulational stability of Stokes waves in deep water. These bifurcations occur repeatedly as the wave steepness grows: new figure-8 bands at speed extrema, their degeneration to vertical slopes, new circular bands at period-doubling points, and reconnections of figure-infinity bands at energy extrema. A sympathetic reader would care because this provides an analytic understanding of how stability changes with amplitude, approaching the highest wave where the profile becomes peaked. The work builds on the analytic theory of Stokes waves and extends the stability analysis using pseudo-differential operators in conformal variables. Numerical computations confirm the normal form predictions for the first two cycles.

Core claim

We derive the criteria and the normal forms for four bifurcations which are repeated recurrently when the steepness of the Stokes wave is increased towards the highest wave with the peaked profile. The four bifurcations are observed in the following order: (a) new figure-8 bands appearing at each extremal point of speed, (b) degeneration of figure-8 bands resulting in vertical slopes, (c) new circular bands around the origin appearing at each period-doubling bifurcation, and (d) reconnection of figure-∞ bands at each extremal point of energy.

What carries the argument

The analytic extension of the modulational stability problem for singular pseudo-differential operators in terms of the Floquet parameter, which allows derivation of normal forms for the recurrent bifurcations.

If this is right

  • The stability spectra of Stokes waves change predictably at specific steepness values corresponding to extrema of speed and energy.
  • Numerical approximations of the spectral bands agree excellently with the normal form theory for the first and second bifurcation cycles.
  • The criteria for bifurcations can be used to predict the onset of instabilities as waves become steeper.
  • Structural assumptions known for Stokes waves enable the analytic treatment of the stability problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the normal forms hold, one could predict the stability near the highest wave without full numerical computation of the spectrum.
  • These bifurcations might relate to the mechanism of wave breaking or the formation of rogue waves in deep water.
  • Extending this to finite depth or other wave types could reveal similar recurrent patterns.

Load-bearing premise

The structural assumptions used in the normal form derivation hold true for the Stokes waves under consideration.

What would settle it

Numerical computation of the stability spectrum at the predicted steepness values for higher bifurcation cycles would falsify the normal forms if the observed band structures deviate from the predicted shapes and slopes.

Figures

Figures reproduced from arXiv: 2604.12269 by Dmitry E. Pelinovsky, Robert Marangell, Sergey Dyachenko.

Figure 1
Figure 1. Figure 1: Transformation of the spectral bands in the modulational stability problem (1.26) when the steepness parameter s of Stokes waves is increased. where a ∈ R is a small parameter of the asymptotic expansions (1.13) and (1.14). The normal form was derived and justified in the previous works [6, 13]. • Panels (b)–(d) show the second bifurcation, for which the slopes of the figure-8 bands become vertical at λ = … view at source ↗
Figure 2
Figure 2. Figure 2: The dependence of H′ (c) (red solid) and ∆(c) (green solid) versus the wave steepness s, with the vertical lines for zeros of c ′ (s) [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The dependence of the first (smallest) eigenvalues of L for co￾periodic (green) and anti-periodic (red) perturbations. for the same eigenvalue λ ∈ C\{0} and is given by vˆ = ∓i⟨1, v⟩η ′ + v, wˆ = ±ic⟨1, v⟩ + w. (2.11) Proof. In the limit µ → 0 ±, we rewrite (1.26) in the explicit form:  Kwˆ = λ (Mvˆ ± i⟨1, vˆ⟩η ′ ), Lvˆ = λ (M∗wˆ − 2cHvˆ ∓ i⟨η ′ ,wˆ⟩ ∓ 2ci⟨1, vˆ⟩). (2.12) Since K1 = 0 and Lη ′ = 0, we hav… view at source ↗
Figure 4
Figure 4. Figure 4: The Newton polytope of cp(λ, ε) = det (A0 + εB + ε 2C − λI). The lower convex hull is the points in the grey triangle. The boundary consists of the lattice points at (0, 2),(2, 1) and (4, 0). where h.o.t. denotes terms of the order of ελ3 , ε 2λ 2 , ε 2λ, and ε 3 . When the coefficients a1 and a2 are nonzero, the lower-convex hull with the boundary connecting (0, 2), (2, 1), and (4, 0) gives the lowest-ord… view at source ↗
Figure 5
Figure 5. Figure 5: The Newton polytope of cp(λ, ε) = det (A0 + εB + ε 2C − λI) when one of the eigenvalues of G1 is zero. The lower convex hull is the points in the grey triangle. The boundary consists of the lattice points at (0, 3),(1, 2),(2, 1) and (4, 0) [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Newton polytope of cp(λ, ε) = det (A0 + εB + ε 2C − λI) with A0 given by (3.16). The lower convex hull is the points in grey. The boundary consists of the lattice points at (0, 2),(2, 1) and (6, 0). 1 2 3 4 5 6 i (λ-power) 2 4 6 8 10 12 j (ϵ-power) [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The same as in [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The same as in [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The Newton polytope of cp(λ, ε) = det (A0 + εB + ε 2C − λI) where A0 is given by (3.23). The lower convex hull is the points in the grey. The boundary consists of the lattice points at (0, 3),(1, 2),(2, 1) and (4, 0) [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The first instance of the hourglass bifurcation, which corresponds to panels (b)–(d) of [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The same as [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The first instance of the ellipse bifurcation, which corresponds to panels (e)–(g) of [PITH_FULL_IMAGE:figures/full_fig_p043_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The same as [PITH_FULL_IMAGE:figures/full_fig_p043_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The first instance of the figure-∞ bifurcation, which corresponds to panels (j)–(l) of [PITH_FULL_IMAGE:figures/full_fig_p048_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The same as [PITH_FULL_IMAGE:figures/full_fig_p048_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The first instance of the figure-8 bifurcation, which correspond to panels (m)–(o) of [PITH_FULL_IMAGE:figures/full_fig_p058_16.png] view at source ↗
read the original abstract

We study the modulational stability problem for the traveling periodic waves (called Stokes waves) in an infinitely deep fluid by using pseudo-differential operators in conformal variables. We derive the criteria and the normal forms for four bifurcations which are repeated recurrently when the steepness of the Stokes wave is increased towards the highest wave with the peaked profile. The four bifurcations are observed in the following order: (a) new figure-8 bands appearing at each extremal point of speed, (b) degeneration of figure-8 bands resulting in vertical slopes, (c) new circular bands around the origin appearing at each period-doubling bifurcation, and (d) reconnection of figure-$\infty$ bands at each extremal point of energy. Our work uses the analytic theory of Stokes waves developed previously for Babenko's equation. The novelty of our work is the analytic extension of the modulational stability problem for singular pseudo-differential operators in terms of the Floquet parameter. The derivation of the normal form uses some structural assumptions which are known to be true for the Stokes waves. For the first and second bifurcation cycles, we compute numerically with a higher-order accuracy the actual values of wave steepness for which the structural assumptions are satisfied and the numerical coefficients of the normal forms to show the excellent agreement between the normal form theory and the numerical approximations of the spectral bands.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives criteria and normal forms for four recurrent bifurcations in the modulational stability spectra of steep Stokes waves in deep fluid, using pseudo-differential operators in conformal variables and the analytic theory of Babenko's equation. The bifurcations occur in sequence as steepness increases toward the limiting peaked wave: (a) new figure-8 bands at speed extrema, (b) degeneration of figure-8 bands to vertical slopes, (c) new circular bands at period-doubling points, and (d) reconnection of figure-∞ bands at energy extrema. High-order numerical computations confirm the steepness values and normal-form coefficients only for the first two cycles, with excellent agreement to the analytic predictions.

Significance. If the structural assumptions hold throughout the steepness range and the analytic extension to singular operators is rigorous, the work offers a systematic normal-form description of recurrent stability changes approaching the highest wave, with potential implications for extreme wave dynamics and breaking. Credit is due for the high-order numerical checks on the first two cycles and for grounding the results in prior Stokes-wave theory.

major comments (2)
  1. [Abstract and numerical results section] The central claim that the four bifurcations recur recurrently up to the highest wave rests on the analytic extension and structural assumptions holding at each cycle. However, the abstract and numerical results section state that high-order computations of steepness values and normal-form coefficients are supplied only for the first and second cycles; no equivalent verification is described for the third or fourth cycles nearer the peaked limit, where operator singularity strengthens. This leaves the recurrence assertion dependent on pattern continuation rather than direct confirmation that assumptions remain valid.
  2. [Derivation of normal forms] § on derivation of normal forms: the criteria and normal forms for bifurcations (a)–(d) are derived under structural assumptions stated to be known for Stokes waves. It is not shown whether these assumptions are proven to persist at higher cycles or merely extrapolated; given that the modulational stability problem involves singular pseudo-differential operators whose properties may degrade near the peaked profile, an explicit check or continuation argument for the third and fourth cycles is load-bearing for the recurrent claim.
minor comments (2)
  1. [Analytic extension section] The presentation of the Floquet-parameter extension for singular operators would benefit from an explicit statement of the domain of analyticity or a reference to the precise theorem in the prior Babenko theory that is being extended.
  2. [Figures] Figure captions for the spectral bands should include the specific steepness values and the order of numerical accuracy used, to allow direct comparison with the normal-form predictions.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the scope of our analytic and numerical results. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and numerical results section] The central claim that the four bifurcations recur recurrently up to the highest wave rests on the analytic extension and structural assumptions holding at each cycle. However, the abstract and numerical results section state that high-order computations of steepness values and normal-form coefficients are supplied only for the first and second cycles; no equivalent verification is described for the third or fourth cycles nearer the peaked limit, where operator singularity strengthens. This leaves the recurrence assertion dependent on pattern continuation rather than direct confirmation that assumptions remain valid.

    Authors: We agree that the manuscript provides high-order numerical verification of steepness values and normal-form coefficients only for the first two bifurcation cycles. The claim of recurrence for subsequent cycles rests on the analytic theory of Babenko's equation and the structural assumptions for Stokes waves, which are known to hold throughout the steepness range up to the limiting wave. The analytic extension of the modulational stability problem is formulated to apply uniformly. We will revise the abstract and numerical results section to state explicitly that numerical confirmation is limited to the first two cycles and that recurrence beyond this is supported by the analytic framework rather than additional direct computations. revision: partial

  2. Referee: [Derivation of normal forms] § on derivation of normal forms: the criteria and normal forms for bifurcations (a)–(d) are derived under structural assumptions stated to be known for Stokes waves. It is not shown whether these assumptions are proven to persist at higher cycles or merely extrapolated; given that the modulational stability problem involves singular pseudo-differential operators whose properties may degrade near the peaked profile, an explicit check or continuation argument for the third and fourth cycles is load-bearing for the recurrent claim.

    Authors: The structural assumptions used in the derivation are established properties of Stokes waves obtained from the analytic theory of Babenko's equation, which applies for the entire family of waves up to the peaked profile. The normal forms are derived under these assumptions via the pseudo-differential operator formulation in conformal variables, with the analytic extension in the Floquet parameter designed to capture the recurrent behavior. We do not supply a separate continuation proof or numerical check for each higher cycle, as the framework is constructed to hold uniformly. We will add a clarifying paragraph in the derivation section noting that persistence of the assumptions at higher cycles follows from the known analytic properties of the Stokes waves rather than case-by-case verification. revision: yes

standing simulated objections not resolved
  • High-order numerical verification of steepness values and normal-form coefficients for the third and fourth bifurcation cycles, which is computationally prohibitive due to operator singularity near the peaked wave.

Circularity Check

0 steps flagged

No significant circularity; analytic derivation with numerical validation for initial cycles

full rationale

The paper derives criteria and normal forms for the four recurrent bifurcations via analytic extension of the modulational stability problem for singular pseudo-differential operators in conformal variables, relying on structural assumptions stated as known for Stokes waves and prior analytic theory for Babenko's equation. It supplies independent high-order numerical verification of steepness values where assumptions hold and of normal-form coefficients, with explicit agreement checks only for the first and second cycles. No step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain whose content is unverified outside the present work; the recurrence claim follows from the analytic continuation rather than from re-labeling inputs. The derivation chain is therefore self-contained against the provided external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on prior analytic theory for Babenko's equation and on structural assumptions for Stokes waves that are taken as known; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Structural assumptions known to be true for the Stokes waves
    Invoked to derive the normal forms for the four bifurcations.

pith-pipeline@v0.9.0 · 5557 in / 1214 out tokens · 25389 ms · 2026-05-10T15:47:49.728406+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    C. J. Amick, L. E. Fraenkel, and J. F. Toland. On the Stokes conjecture for the wave of extreme form. Acta Mathematica, 148(1):193–214, 1982

    work page 1982

  2. [2]

    K. I. Babenko. Some remarks on the theory of surface waves of finite amplitude. InDoklady Akademii Nauk, volume 294, pages 1033–1037. Russian Academy of Sciences, 1987

    work page 1987

  3. [3]

    A. M. Balk. A Lagrangian for water waves.Physics of Fluids, 8:416–420, 1996

    work page 1996

  4. [4]

    T. B. Benjamin and J. E. Feir. The disintegration of wave trains on deep water.Journal of Fluid Mechanics, 27(3):417–430, 1967

    work page 1967

  5. [5]

    T. B. Benjamin and P. J. Olver. Hamiltonian structure, symmetries and conservation laws for water waves.Journal of Fluid Mechanics, 125:137–185, 1982

    work page 1982

  6. [6]

    Berti, A

    M. Berti, A. Maspero, and P. Ventura. Full description of Benjamin-Feir instability of Stokes waves in deep water.Inventiones mathematicae, 230(2):651–711, 2022

    work page 2022

  7. [7]

    Berti, A

    M. Berti, A. Maspero, and P. Ventura. Benjamin-Feir instability of Stokes waves in finite depth.Arch. Rational Mech. Anal., 247:91, 2023

    work page 2023

  8. [8]

    Bridges and A

    T. Bridges and A. Mielke. A proof of the Benjamin-Feir instability.Archive for rational mechanics and analysis, 133:145–198, 1995

    work page 1995

  9. [9]

    J. C. Bronski, V. M. Hur, and R. Marangell. Floquet theory and stability analysis for Hamiltonian PDEs.Nonlinearity, 37:125010 (42 pages), 2024. 62 S. DYACHENKO, R. MARANGELL, AND D. E. PELINOVSKY

    work page 2024

  10. [10]

    Buffoni, E

    B. Buffoni, E. N. Dancer, and J. F. Toland. The regularity and local bifurcation of steady periodic water waves.Arch. Rational Mech. Anal., 152:207–240, 2000

    work page 2000

  11. [11]

    Buffoni, E

    B. Buffoni, E. N. Dancer, and J. F. Toland. The sub-harmonic bifurcations of Stokes waves.Arch. Rational Mech. Anal., 152:241–271, 2000

    work page 2000

  12. [12]

    Creedon, B

    R. Creedon, B. Deconinck, and O. Trichtchenko. High-frequency instabilities of a Boussinesq–Whitham system: a perturbative approach.Fluids, 6(4):136, 2021

    work page 2021

  13. [13]

    R. P. Creedon and B. Deconinck. A high-order asymptotic analysis of the Benjamin–Feir instability spectrum in arbitrary depth.Journal of Fluid Mechanics, 956:A29, 2023

    work page 2023

  14. [14]

    Cui and D

    S. Cui and D. E. Pelinovsky. Instability bands for periodic traveling waves in the modified Korteweg-de Vries equation.Proc. R. Soc. A, 481:20240993 (18 pages), 2025

    work page 2025

  15. [15]

    Deconinck, S

    B. Deconinck, S. A. Dyachenko, P. M. Lushnikov, and A. Semenova. The dominant instability of near- extreme Stokes waves.Proceedings of the National Academy of Sciences, 120(32):e2308935120, 2023

    work page 2023

  16. [16]

    Deconinck, S

    B. Deconinck, S. A. Dyachenko, and A. Semenova. Self-similarity and recurrence in stability spectra of near-extreme Stokes waves.Journal of Fluid Mechanics, 995:A2, 2024

    work page 2024

  17. [17]

    Deconinck and K

    B. Deconinck and K. Oliveras. The instability of periodic surface gravity waves.Journal of Fluid Me- chanics, 675:141–167, 2011

    work page 2011

  18. [18]

    A. I. Dyachenko, E. A. Kuznetsov, M. D. Spector, and V. E. Zakharov. Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping).Physics Letters A, 221(1-2):73–79, 1996

    work page 1996

  19. [19]

    Dyachenko and D

    S. Dyachenko and D. E. Pelinovsky. Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy.Physica D, 483:134925, 2025

    work page 2025

  20. [20]

    S. A. Dyachenko, V. M. Hur, and D. A. Silantyev. Almost extreme waves.Journal of Fluid Mechanics, 955:A17, 2023

    work page 2023

  21. [21]

    S. A. Dyachenko, P. M. Lushnikov, and A. O. Korotkevich. Branch cuts of Stokes wave on deep water. Part I: numerical solution and Pad´ e approximation.Studies in Applied Mathematics, 137(4):419–472, 2016

    work page 2016

  22. [22]

    S. A. Dyachenko and A. Semenova. Canonical conformal variables based method for stability of Stokes waves.Studies in Applied Mathematics, 150(3):705–715, 2023

    work page 2023

  23. [23]

    S. A. Dyachenko and A. Semenova. Quasiperiodic perturbations of Stokes waves: Secondary bifurcations and stability.Journal of Computational Physics, 492:112411, 2023

    work page 2023

  24. [24]

    E. J. Hinch.Perturbation Methods. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, 1991

    work page 1991

  25. [25]

    Hormander.An introduction to complex analysis in several variables

    L. Hormander.An introduction to complex analysis in several variables. Elsevier, 1973

    work page 1973

  26. [26]

    V. M. Hur and Z. Yang. Unstable Stokes waves.Arch. Rational Mech. Anal., 247:62, 2023

    work page 2023

  27. [27]

    F. C. Kirwan.Complex algebraic curves. Cambridge University Press, 1992

    work page 1992

  28. [28]

    A. O. Korotkevich, P. M. Lushnikov, A. Semenova, and S. A. Dyachenko. Superharmonic instability of Stokes waves.Studies in Applied Mathematics, 150(1):119–134, 2023

    work page 2023

  29. [29]

    Levi-Civita

    T. Levi-Civita. D´ etermination rigoureuse des ondes permanentes d’ampleur finie.Mathematische An- nalen, 93(1):264–314, 1925

    work page 1925

  30. [30]

    Locke and D

    S. Locke and D. E. Pelinovsky. On smooth and peaked traveling waves in a local model for shallow water waves.Journal of Fluid Mechanics, 1004:A1, 2025

    work page 2025

  31. [31]

    Locke and D

    S. Locke and D. E. Pelinovsky. Peaked Stokes waves as solutions of Babenko’s equation.Applied Math- ematics Letters, 161:109359, 2025

    work page 2025

  32. [32]

    Longuet-Higgins and M

    M. Longuet-Higgins and M. Tanaka. On the crest instabilities of steep surface waves.Journal of Fluid Mechanics, 336:51–68, 1997

    work page 1997

  33. [33]

    M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water I. Su- perharmonics.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 360(1703):471–488, 1978. BIFURCATIONS OF STABILITY SPECTRA FOR STEEP STOKES W A VES 63

    work page 1978

  34. [34]

    M. S. Longuet-Higgins and M. J. H. Fox. Theory of the almost–highest wave. Part 2. Matching and analytic extension.Journal of Fluid Mechanics, 85:769–786, 1978

    work page 1978

  35. [35]

    Ma and A

    Y. Ma and A. Edelman. Nongeneric Eigenvalue Perturbations of Jordan Blocks.Linear Algebra and Applications, 273:45–63, 1998

    work page 1998

  36. [36]

    J. Moro, J. V. Burke, and M. L. Overton. On the Lidskii–Vishik–Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure.SIAM Journal on Matrix Analysis and Appli- cations, 18(4):793–817, 1997

    work page 1997

  37. [37]

    Murashige and W

    S. Murashige and W. Choi. Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping.Journal of Fluid Mechanics, 885, 2020

    work page 2020

  38. [38]

    A. I. Nekrasov. On waves of permanent type.Izv. Ivanovo-Voznesensk. Polite. Inst., 3:52–65, 1921

    work page 1921

  39. [39]

    H. Q. Nguyen and W. A. Strauss. Proof of modulational instability of Stokes waves in deep water. Communications on Pure and Applied Mathematics, 76(5):1035–1084, 2023

    work page 2023

  40. [40]

    P. I. Plotnikov. Nonuniqueness of solutions of the problem of solitary waves and bifurcation of critical points of smooth functionals.Math USSR Izvestiya, 38:333–357, 1992

    work page 1992

  41. [41]

    P. I. Plotnikov. A proof of the Stokes conjecture in the theory of surface waves.Studies in Applied Mathematics, 108(2):217–244, 2002

    work page 2002

  42. [42]

    P. G. Saffman. The superharmonic instability of finite-amplitude water waves.J. Fluid Mech., 159:169– 174, 1985

    work page 1985

  43. [43]

    Two-crested Stokes waves.Applied Mathematics Letters, 167:109560, 2025

    A Semenova. Two-crested Stokes waves.Applied Mathematics Letters, 167:109560, 2025

    work page 2025

  44. [44]

    G. G. Stokes. On the theory of oscillatory waves.Transactions of the Cambridge Philosophical Society, 8:441, 1847

    work page

  45. [45]

    G. G. Stokes. On the theory of oscillatory waves.Mathematical and Physical Papers, 1:197, 1880

    work page

  46. [46]

    D. Struik. D´ etermination rigoureuse des ondes irrotationelles p´ eriodiques dans un canal ¨ a profondeur finie.Math. Ann., 95:595–634, 1926

    work page 1926

  47. [47]

    M. Tanaka. The stability of steep gravity waves.Journal of the physical society of Japan, 52(9):3047– 3055, 1983

    work page 1983

  48. [48]

    M. Tanaka. The stability of steep gravity waves. ii.J. Fluid Mech., 156:281–289, 1985

    work page 1985

  49. [49]

    J. F. Toland. On the existence of a wave of greatest height and Stokes’s conjecture.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 363(1715):469–485, 1978

    work page 1978

  50. [50]

    V. E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid.Journal of Applied Mechanics and Technical Physics, 9(2):190–194, 1968. (S. Dyachenko)Department of Mathematics, State University of New York at Buffalo, Buffalo, New York, USA, 14260 Email address:sergeydy@buffalo.edu (R. Marangell)School of Mathematics and Sta...

    work page 1968