We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size $$ \varepsilon $$in a sufficiently smooth Sobolev space leads to a solution that remains in an $$ \varepsilon $$-ball of the same Sobolev space up times of order $$ \varepsilon ^{-2}$$. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.

VL - 3 SN - 2523-3688 UR - https://doi.org/10.1007/s42286-020-00036-8 IS - 1 JO - Water Waves ER - TY - JOUR T1 - Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential JF - Journal of the European Mathematical Society Y1 - 2013 A1 - Massimiliano Berti A1 - Philippe Bolle AB - We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td , d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C∞ then the solutions are C∞. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates. © European Mathematical Society 2013. VL - 15 N1 - cited By (since 1996)5 ER - TY - JOUR T1 - Quasi-periodic solutions of completely resonant forced wave equations JF - Comm. Partial Differential Equations 31 (2006) 959 - 985 Y1 - 2006 A1 - Massimiliano Berti A1 - Michela Procesi AB - We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number. UR - http://hdl.handle.net/1963/2234 U1 - 2010 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Quasi-periodic oscillations for wave equations under periodic forcing JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 Y1 - 2005 A1 - Massimiliano Berti A1 - Michela Procesi PB - Accademia Nazionale dei Lincei UR - http://hdl.handle.net/1963/4583 U1 - 4350 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER -