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.

1 aBerti, Massimiliano1 aFeola, Roberto1 aFranzoi, Luca uhttps://doi.org/10.1007/s42286-020-00036-801345nas a2200145 4500008004100000022001300041245009800054210006900152300001200221490000700233520079700240100002401037700002001061856011801081 2013 eng d a1435985500aQuasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential0 aQuasiperiodic solutions with Sobolev regularity of NLS on Td wit a229-2860 v153 aWe 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 2013.1 aBerti, Massimiliano1 aBolle, Philippe uhttps://www.math.sissa.it/publication/quasi-periodic-solutions-sobolev-regularity-nls-td-multiplicative-potential00676nas a2200109 4500008004300000245007400043210006900117520029900186100002400485700002100509856003600530 2006 en_Ud 00aQuasi-periodic solutions of completely resonant forced wave equations0 aQuasiperiodic solutions of completely resonant forced wave equat3 aWe 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.1 aBerti, Massimiliano1 aProcesi, Michela uhttp://hdl.handle.net/1963/223400410nas a2200109 4500008004100000245007400041210006900115260003500184100002400219700002100243856003600264 2005 en d00aQuasi-periodic oscillations for wave equations under periodic forcing0 aQuasiperiodic oscillations for wave equations under periodic for bAccademia Nazionale dei Lincei1 aBerti, Massimiliano1 aProcesi, Michela uhttp://hdl.handle.net/1963/4583