We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This proves a conjecture of Zakharov-Dyachenko [55] suggested by the formal Birkhoff integrability of the water waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order ε−3. This time scale is expected to be optimal. Main difficulties in the proof are the presence of non-trivial resonant four-waves interactions, the so-called Benjamin-Feir resonances, the small divisors arising from near-resonances and the quasilinear nature of the equations. Some of the main ingredients that we use are: (1) a reduction procedure to constant coefficient operators up to smoothing remainders, that, together with the verification of key algebraic cancellations of the system, implies the integrability of the equations at non-negative orders; (2) a Poincaré-Birkhoff normal form of the smoothing remainders that deals with near-resonances; (3) an a priori algebraic identification argument of the above Poincaré- Birkhoff normal form equations with the formal Hamiltonian computations of [55, 19, 27, 17], that allows us to handle the Benjamin-Feir resonances.

Birkhoff normal form and long time existence for periodic gravity water waves / Berti, Massimiliano; Feola, Roberto; Pusateri, Fabio. - In: COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS. - ISSN 0010-3640. - (2022), pp. 1-79. [10.1002/cpa.22041]

Birkhoff normal form and long time existence for periodic gravity water waves

Berti, Massimiliano;Feola, Roberto;Pusateri, Fabio
2022

Abstract

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This proves a conjecture of Zakharov-Dyachenko [55] suggested by the formal Birkhoff integrability of the water waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order ε−3. This time scale is expected to be optimal. Main difficulties in the proof are the presence of non-trivial resonant four-waves interactions, the so-called Benjamin-Feir resonances, the small divisors arising from near-resonances and the quasilinear nature of the equations. Some of the main ingredients that we use are: (1) a reduction procedure to constant coefficient operators up to smoothing remainders, that, together with the verification of key algebraic cancellations of the system, implies the integrability of the equations at non-negative orders; (2) a Poincaré-Birkhoff normal form of the smoothing remainders that deals with near-resonances; (3) an a priori algebraic identification argument of the above Poincaré- Birkhoff normal form equations with the formal Hamiltonian computations of [55, 19, 27, 17], that allows us to handle the Benjamin-Feir resonances.
1
79
10.1002/cpa.22041
https://arxiv.org/abs/1810.11549
Berti, Massimiliano; Feola, Roberto; Pusateri, Fabio
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/117051
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