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Full Description
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 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 the combined problems of small divisors and the fully-nonlinear nature of the equations.
The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known "Benjamin-Feir resonances". We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.
Contents
Chapters
1. Introduction
2. Functional setting
3. Normal forms and integrability properties of the pure gravity water waves
4. Weak Birkhoff normal form
5. The nonlinear functional setting
6. Approximate inverse
7. The linearized operator in the normal directions
8. Symmetrization of the linearized operator at the highest order
9. Block-diagonalization
10. Reduction at the highest orders
11. Linear Birkhoff normal form
12. Inversion of the linearized operator
13. The Nash-Moser nonlinear iteration
A. Flows and conjugations
B. Technical lemmata
