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}
}
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
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
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
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?
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
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
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
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
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
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
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