# Quasi-periodic traveling waves on an infinitely deep fluid under gravity

@article{Feola2020QuasiperiodicTW,
title={Quasi-periodic traveling waves on an infinitely deep fluid under gravity},
author={Roberto Feola and Filippo Giuliani},
journal={arXiv: Analysis of PDEs},
year={2020}
}
• Published 17 May 2020
• Mathematics, Physics
• arXiv: Analysis of PDEs
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a \emph{completely resonant} elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with…
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#### References

SHOWING 1-10 OF 38 REFERENCES
KAM for autonomous quasi-linear perturbations of KdV
• Mathematics
• 2014
Abstract We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian differentiable
Nash-Moser Theory for Standing Water Waves
• Mathematics
• 2001
Abstract We consider a perfect fluid in periodic motion between parallel vertical walls, above a horizontal bottom and beneath a free boundary at constant atmospheric pressure. Gravity acts
An integrable normal form for water waves in infinite depth
• Mathematics
• 1995
Abstract We consider the Birkhoff normal form for the water wave problem posed in a fluid of infinite depth, with the starting point of our analysis a version of the Hamiltonian given by Zakharov. We
Is free-surface hydrodynamics an integrable system?
• Physics
• 1994
Abstract A strong argument is found in support of the integrability of free-surface hydrodynamics in the one-dimensional case. It is shown that the first term in the perturbation series in powers of
Birkhoff Normal form for Gravity Water Waves
• Physics
• 2020
We consider the gravity water waves system with a one-dimensional periodic interface in infinite depth, and present the proof of the rigorous reduction of these equations to their cubic Birkhoff
Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
• Mathematics
Annales de l'Institut Henri Poincaré C, Analyse non linéaire
• 2019
We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schr\"odinger equations on the circle. After a paralinearization of the equation, we perform
Long-time existence for multi-dimensional periodic water waves
• Mathematics, Physics
Geometric and Functional Analysis
• 2019
We prove an extended lifespan result for the full gravity-capillary water waves system with a 2 dimensional periodic interface: for initial data of sufficiently small size $${\varepsilon}$$ε, smooth
On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances
• Mathematics, Physics
Journal of Differential Equations
• 2019
We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$ with $\mathtt{c}\neq 0$. In \cite{Deg} the authors proved
Global Regularity for 2d Water Waves with Surface Tension
• Mathematics
Memoirs of the American Mathematical Society
• 2018
We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for