Almost global large deviations principle for the KdV equation

Ricardo Grande; Riccardo Berforini D'Aquino

arxiv: 2605.00484 · v1 · submitted 2026-05-01 · 🧮 math.AP · math-ph· math.MP· math.PR

Almost global large deviations principle for the KdV equation

Riccardo Berforini D'Aquino , Ricardo Grande This is my paper

Pith reviewed 2026-05-09 19:27 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPmath.PR

keywords Korteweg-de Vries equationlarge deviations principleextreme wavesBirkhoff normal formintegrable systemsphase synchronizationweakly nonlinear regime

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The pith

A large deviations principle governs the probability of extreme amplitudes for KdV solutions over polynomial timescales with small random data.

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

The paper establishes a large deviations principle for the supremum of solutions to the Korteweg-de Vries equation on the torus with random initial data of average size ε. This principle holds uniformly for times up to ε to any fixed negative integer power and supplies the leading-order exponential asymptotics for the probability of unusually large waves. Because the equation is integrable, the Fourier moduli remain nearly conserved, eliminating resonant energy transfer as a route to extremes. Large amplitudes therefore require quasi-synchronization of phases, and the authors show this mechanism dominates in the weakly nonlinear regime. Their proof combines a Birkhoff normal form reduction with probabilistic estimates that exploit the stability of the integrable tori to control alignment probabilities over long times.

Core claim

We prove an almost-global large deviations principle for the supremum of the KdV solution on the torus, for initial data of size ε, that holds for all times t ≤ ε^{-n} with n any fixed natural number. The associated rate function gives the leading asymptotics of the probability of observing amplitudes much larger than ε. In this integrable setting the dynamics stay close to invariant tori on which the Fourier moduli are almost conserved, so extreme events can arise only through coherent structures or dispersive focusing realized by phase quasi-synchronization. Birkhoff normal form stability is used to bound the probability of such synchronization over the entire polynomial time interval.

What carries the argument

Birkhoff normal form analysis that preserves the stability of the integrable dynamics and controls the probability of phase quasi-synchronization over polynomial timescales.

Load-bearing premise

The stability of the integrable dynamics under the Birkhoff normal form remains sufficient to control the probability of phase quasi-synchronization over the full polynomial timescale t ≤ ε^{-n}.

What would settle it

A direct computation or simulation that shows either the probability of phase alignment exceeds the normal-form upper bound or the observed large-amplitude probabilities deviate from the predicted large-deviation rate at times scaling as ε^{-n} for large n.

read the original abstract

We study extreme wave formation for the Korteweg-de Vries equation on the torus with random initial data of average size $\epsilon$. We establish a large deviations principle for the supremum of the solution over arbitrarily long polynomial timescales $t \leq \epsilon^{-n}$ for any fixed natural number $n$. This identifies the leading-order asymptotics of the probability of observing unusually large amplitudes. In this integrable setting, the dynamics evolves on invariant tori where Fourier moduli are almost conserved, ruling out mechanisms for extreme wave formation based on resonant energy exchange. As a result, large amplitudes can only arise through coherent structures or dispersive focusing, which corresponds to the quasi-synchronization of many phases. We show that the latter is dominant in the weakly nonlinear regime. Our approach combines a Birkhoff normal form analysis with probabilistic arguments, exploiting the stability of the integrable dynamics to control the probability of phase quasi-synchronization over long timescales.

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.

Desk Editor's Note private letter to a colleague

The KdV paper claims an almost-global LDP for extreme amplitudes via normal forms and phase sync, but the long-time remainder control looks like the part that needs the most checking.

read the letter

The key point is that the authors prove a large deviations principle for unusually large amplitudes in the KdV equation that holds for times as long as any fixed power of 1 over epsilon, by reducing the problem to the probability of phase quasi-synchronization using Birkhoff normal forms. This is new because prior work on large deviations for integrable PDEs has mostly been local in time or for shorter scales, and combining the normal form stability with probabilistic estimates to get almost global control appears fresh. The paper does well by clearly using the conserved quantities from integrability to rule out resonant mechanisms for wave growth, leaving only dispersive focusing from phase alignment as the way to get extremes. That separation is helpful and lets them focus the probabilistic part on the right quantity. The main soft spot is the accumulation of the normal form error over the long times. The Birkhoff normal form gives a small remainder, but integrating that over intervals of length epsilon to the minus n can produce a phase drift of size epsilon to some power that depends on n. For the large deviation rate to be accurate, this drift has to stay small enough not to affect the synchronization probabilities in the tail events. The abstract does not detail how the normal form order or the stability constants are chosen to make this work uniformly for every n, so it is possible that the bounds only close for small enough n or with n-dependent adjustments that are not stated. The stress test concern about the remainder invalidating the probability bounds seems worth checking in the full argument. If the perturbation changes the effective frequencies or the condition for quasi-synchronization, the leading order asymptotics could be off. This paper targets mathematicians working on nonlinear wave equations, integrable systems, and applications of large deviation theory to PDEs. Someone studying rigorous models for rogue waves or long-time behavior in dispersive equations would find the approach useful. It deserves serious peer review because the claim is specific and the strategy is coherent on paper, even though the long-time estimates require verification. I would recommend sending it to referees rather than desk rejecting it.

Referee Report

1 major / 2 minor

Summary. The paper claims to prove an almost-global large-deviations principle for the supremum of solutions to the periodic KdV equation with random initial data of size ε. The LDP holds uniformly for times up to ε^{-n} for any fixed natural number n and identifies the leading-order probability of large amplitudes as arising from quasi-synchronization of phases on the invariant tori furnished by a Birkhoff normal form, rather than from resonant energy exchange.

Significance. If the central estimates close, the result supplies a rigorous probabilistic description of extreme-wave formation in an integrable dispersive PDE on polynomial timescales, separating the contribution of dispersive focusing from resonant mechanisms. This is a concrete advance for the study of large deviations in Hamiltonian PDEs with random data and may serve as a template for other integrable models.

major comments (1)

[§4.3] §4.3 (normal-form remainder and phase evolution): The Birkhoff normal-form remainder is controlled in a suitable norm by O(ε^m) for some m, yet the manuscript does not state how m or the implicit constants depend on the arbitrary fixed n. Over intervals of length ε^{-n} the integrated phase drift is then of size ε^{m-n}; for the quasi-synchronization probability bounds (and hence the LDP rate) to remain valid uniformly in the large-deviation regime, this drift must be o(1) for every n. An explicit scaling or a uniform-in-n stability statement is required to close the argument.

minor comments (2)

[Theorem 1.1] The statement of the LDP (Theorem 1.1) should specify the precise topology on the space of measures or functions in which the large-deviation principle is asserted.
[§2 and §5] Notation for the action-angle variables and the frequency map is introduced in §2 but reused without re-statement in the probabilistic estimates of §5; a short reminder table would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this key point about the dependence of the normal-form estimates on the time-scale parameter n. We agree that the manuscript should make this dependence explicit to close the argument, and we will revise accordingly.

read point-by-point responses

Referee: [§4.3] §4.3 (normal-form remainder and phase evolution): The Birkhoff normal-form remainder is controlled in a suitable norm by O(ε^m) for some m, yet the manuscript does not state how m or the implicit constants depend on the arbitrary fixed n. Over intervals of length ε^{-n} the integrated phase drift is then of size ε^{m-n}; for the quasi-synchronization probability bounds (and hence the LDP rate) to remain valid uniformly in the large-deviation regime, this drift must be o(1) for every n. An explicit scaling or a uniform-in-n stability statement is required to close the argument.

Authors: We agree that the current presentation does not explicitly address the dependence of m on n. In the Birkhoff normal-form analysis, the order of the normal form can be taken arbitrarily high. For each fixed natural number n we choose the normal-form order large enough that m = m(n) satisfies m > n + 1 (for instance), so that the integrated phase drift over [0, ε^{-n}] is O(ε^{m-n}) = O(ε) which is o(1) as ε → 0. The implicit constants in the remainder estimates depend on n only through this choice of m(n); because the large-deviations statement is formulated separately for each fixed n, such n-dependent constants are permissible. We will add a clarifying paragraph in §4.3 stating this scaling explicitly and verifying that the quasi-synchronization probability bounds continue to hold with the required o(1) drift. This completes the argument without needing a uniform-in-n stability result. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent Birkhoff normal forms and external probabilistic LDP tools

full rationale

The paper's central claim combines a Birkhoff normal form reduction (standard in integrable PDEs, with stability estimates independent of the target LDP) with separate probabilistic large-deviation estimates on phase synchronization. No step defines the probability of large amplitudes in terms of itself, fits a parameter to the output quantity, or reduces the result to a self-citation chain. The normal-form remainder is controlled by external estimates whose scaling is stated to suffice for any fixed polynomial time; this is a technical assumption, not a definitional loop. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and long-time stability of a Birkhoff normal form for the KdV equation together with standard probabilistic large-deviation estimates; no free parameters or new entities are introduced.

axioms (1)

domain assumption KdV admits a Birkhoff normal form that preserves the integrable torus structure and controls the dynamics over polynomial times t ≤ ε^{-n}
Invoked to rule out resonant energy exchange and to reduce extreme-wave formation to phase quasi-synchronization.

pith-pipeline@v0.9.0 · 5458 in / 1298 out tokens · 44049 ms · 2026-05-09T19:27:32.672257+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

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R. Grande, and Z. Hani , Rigorous derivation of damped-driven wave turbulence theory , Arch. Ration. Mech. Anal. 250, 27 (2026)

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R. Grande, K.M. Kurianski, and G. Staffilani , On the nonlinear Dysthe equation , Nonlinear Anal. 207 , 112292 (2021)

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D.J. Korteweg, and G. de Vries , On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , Philos. Mag. 39 (240) , 422--443 (1895)

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M.D. Kruskal, R.M. Miura, C.S. Gardner, and N.J. Zabusky , Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws , J. Math. Phys. 11 , 952--960 (1970)

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Lax , Periodic solutions of the KdV equation , Comm

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Liang, and Y

R. Liang, and Y. Wang , Large deviations principle for the cubic NLS equation with slowly decaying data , preprint arXiv:2512.07773 (2025)

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Magri , A simple model of the integrable Hamiltonian equation , J

F. Magri , A simple model of the integrable Hamiltonian equation , J. Math. Phys. 19 (5), 1156--1162 (1978)

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R. Mosincat, D. Pilod, and J.-C. Saut , Global well-posedness and scattering for the Dysthe equation in L^2( ^2) , J. Math. Pures Appl. 149 (9), pp. 73–97 (2021)

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[1] [1]

Ardhuin, T

F. Ardhuin, T. Postec, M. Accensi, J. Piolle, G. Dodet, M. Passaro, M. De Carlo, R. Husson, G. Guitton, and F. Collard , Sizing the largest ocean waves using the SWOT mission , Proc. Natl. Acad. Sci. U.S.A. 122 (38) e2513381122 (2025)

work page 2025

[2] [2]

Arnold , Mathematical methods of classical mechanics , Springer, 2013

V.I. Arnold , Mathematical methods of classical mechanics , Springer, 2013

work page 2013

[3] [3]

Babin, A.A

A.V. Babin, A.A. Ilyin, and E.S. Titi , On the regularization mechanism for the periodic Korteweg–de Vries equation , Comm. Pure Appl. Math. 64 (5), 591--648 (2011)

work page 2011

[4] [4]

Bernier, and B

J. Bernier, and B. Grébert , Long time dynamics for generalized Korteweg–de Vries and Benjamin–Ono equations , Arch. Ration. Mech. Anal. 241 (3), 1139--1241 (2021)

work page 2021

[5] [5]

Berti, R

M. Berti, R. Grande, A. Maspero, and G. Staffilani , Rogue waves and large deviations for 2D pure gravity deep water waves , arXiv:2510.15159 https://arxiv.org/abs/2510.15159 (2025)

work page arXiv 2025

[6] [6]

Bharucha-Reid , Fixed point theorems in probabilistic analysis , Bulletin of the American Mathematical Society 82 (5), 641--657 (1976)

A.T. Bharucha-Reid , Fixed point theorems in probabilistic analysis , Bulletin of the American Mathematical Society 82 (5), 641--657 (1976)

work page 1976

[7] [7]

Bochnak, and J

J. Bochnak, and J. Siciak , Polynomials and multilinear mappings in topological vector-spaces , Studia Mathematica 39 (1), 59--76 (1971)

work page 1971

[8] [8]

Castaing, and M

C. Castaing, and M. Valadier , Convex Analysis and Measurable Multifunctions , Lecture Notes in Mathematics, Springer Berlin, Heidelberg, 1977

work page 1977

[9] [9]

Christou, and K

M. Christou, and K. Ewans , Field measurements of rogue water waves , J. of Physical Oceanography 44, 2317–2335 (2014)

work page 2014

[10] [10]

Colliander, M

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao Sharp Global well-posedness for KdV and modified KdV on and , J. Amer. Math. Soc. 16 (3), 705--749 (2023)

work page 2023

[11] [11]

Dematteis, T

G. Dematteis, T. Grafke, M. Onorato, and E. Vanden-Eijnden , Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves , Phys. Rev. X 9 (4), 041057 (2019)

work page 2019

[12] [12]

Dematteis, T

G. Dematteis, T. Grafke, and E. Vanden-Eijnden , Rogue waves and large deviations in deep sea , Proc. Natl. Acad. Sci. USA 115 (5), 855--860 (2018)

work page 2018

[13] [13]

Deng, and Z

Y. Deng, and Z. Hani , Full derivation of the wave kinetic equation , Invent. Math. 233 (2), 543–724 (2023)

work page 2023

[14] [14]

Y. Deng, A. Ionescu, and F. Pusateri , On the wave turbulence theory of 2D gravity waves, I: deterministic energy estimates, Comm. Pure Appl. Math. 78 (2), 211–322 (2025)

work page 2025

[15] [15]

Erdo g an, and N

M.B. Erdo g an, and N. Tzirakis , Dispersive partial differential equations. Wellposedness and applications , London Math. Soc. Student Texts, 86 , Cambridge Univ. Press, Cambridge, 2016

work page 2016

[16] [16]

Fasano, and S

A. Fasano, and S. Marmi , Analytical mechanics , Oxford Graduate Texts, Oxford Univ. Press, 2010

work page 2010

[17] [17]

Feola, F

R. Feola, F. Giuliani and S. Pasquali , On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances , Journal of Differential Equations 266 (6), 3390--3437 (2019)

work page 2019

[18] [18]

Feola, F

R. Feola, F. Giuliani and M. Procesi , Reducible KAM Tori for the Degasperis–Procesi Equation , Communications in Mathematical Physics 377 (3), 1681--1759 (2020)

work page 2020

[19] [19]

Gardner , Korteweg–de Vries equation and generalizations

C.S. Gardner , Korteweg–de Vries equation and generalizations. IV. The Korteweg–de Vries equation as a Hamiltonian system , J. Math. Phys. 12 (8), 1548--1551 (1971)

work page 1971

[20] [20]

Gesztesy, and H

F. Gesztesy, and H. Holden , Soliton Equations and Their Algebro-Geometric Solutions , Cambridge University Press, June 2003

work page 2003

[21] [21]

Garrido, R

M.A. Garrido, R. Grande, K.M. Kurianski, and G. Staffilani , Large deviations principle for the cubic NLS equation , Comm. Pure Appl. Math. 76 (12), 4087--4136 (2023)

work page 2023

[22] [22]

Grande , Resonant large deviations principle for the beating NLS equation , SIAM J

R. Grande , Resonant large deviations principle for the beating NLS equation , SIAM J. Math. Anal. 57 (6), 6598-6632 (2025)

work page 2025

[23] [23]

Grande, and Z

R. Grande, and Z. Hani , Rigorous derivation of damped-driven wave turbulence theory , Arch. Ration. Mech. Anal. 250, 27 (2026)

work page 2026

[24] [24]

Grande, K.M

R. Grande, K.M. Kurianski, and G. Staffilani , On the nonlinear Dysthe equation , Nonlinear Anal. 207 , 112292 (2021)

work page 2021

[25] [25]

Hasselmann, D

K. Hasselmann, D. Olbers , Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP) Reihe A, Erg\"anzung zur Deut Hydrogr Z 1–95 (1973)

work page 1973

[26] [26]

Kappeler, and J

T. Kappeler, and J. Pöschel , KdV & KAM , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer, Berlin–Heidelberg, 2003

work page 2003

[27] [27]

Korteweg, and G

D.J. Korteweg, and G. de Vries , On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , Philos. Mag. 39 (240) , 422--443 (1895)

work page

[28] [28]

Kruskal, R.M

M.D. Kruskal, R.M. Miura, C.S. Gardner, and N.J. Zabusky , Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws , J. Math. Phys. 11 , 952--960 (1970)

work page 1970

[29] [29]

Lax , Almost periodic solutions of the KdV equation , SIAM Rev

P.D. Lax , Almost periodic solutions of the KdV equation , SIAM Rev. 18 (3), 351--375 (1976)

work page 1976

[30] [30]

Lax , Periodic solutions of the KdV equation , Comm

P.D. Lax , Periodic solutions of the KdV equation , Comm. Pure Appl. Math. 28 (1), 141--188 (1975)

work page 1975

[31] [31]

Liang, and Y

R. Liang, and Y. Wang , Large deviations principle for the cubic NLS equation with slowly decaying data , preprint arXiv:2512.07773 (2025)

work page arXiv 2025

[32] [32]

Magri , A simple model of the integrable Hamiltonian equation , J

F. Magri , A simple model of the integrable Hamiltonian equation , J. Math. Phys. 19 (5), 1156--1162 (1978)

work page 1978

[33] [33]

Miura, C.S

R.M. Miura, C.S. Gardner, and M.D. Kruskal , Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion , J. Math. Phys. 9 (8), 1204--1209 (1968)

work page 1968

[34] [34]

Mosincat, D

R. Mosincat, D. Pilod, and J.-C. Saut , Global well-posedness and scattering for the Dysthe equation in L^2( ^2) , J. Math. Pures Appl. 149 (9), pp. 73–97 (2021)

work page 2021

[35] [35]

Oh, and N

T. Oh, and N. Tzvetkov , Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation , Probab. Theory Relat. Fields 169, 1121-1168 (2017)

work page 2017

[36] [36]

Onorato, S

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F.T. Arecchi , Rogue waves and their generating mechanisms in different physical contexts , Phys. Rep. 528 (2), 47--89 (2013)

work page 2013

[37] [37]

B. J. Pettis , On Integration in Vector Spaces , Trans. Amer. Math. Soc. 44 (2), 277--304 (1938)

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