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.

%V 3 %P 85 - 115 %8 2021/04/01 %@ 2523-3688 %G eng %U https://doi.org/10.1007/s42286-020-00036-8 %N 1 %! Water Waves %0 Journal Article %J Journal of the European Mathematical Society %D 2013 %T Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential %A Massimiliano Berti %A Philippe Bolle %X 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. %B Journal of the European Mathematical Society %V 15 %P 229-286 %G eng %R 10.4171/JEMS/361 %0 Journal Article %J Comm. Partial Differential Equations 31 (2006) 959 - 985 %D 2006 %T Quasi-periodic solutions of completely resonant forced wave equations %A Massimiliano Berti %A Michela Procesi %X 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. %B Comm. Partial Differential Equations 31 (2006) 959 - 985 %G en_US %U http://hdl.handle.net/1963/2234 %1 2010 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-16T07:48:45Z\\nNo. of bitstreams: 1\\n0504406v1.pdf: 330239 bytes, checksum: 5dbf59bdd590a6876ea206f70cf0ecc9 (MD5) %R 10.1080/03605300500358129 %0 Journal Article %J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %D 2005 %T Quasi-periodic oscillations for wave equations under periodic forcing %A Massimiliano Berti %A Michela Procesi %B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %I Accademia Nazionale dei Lincei %G en %U http://hdl.handle.net/1963/4583 %1 4350 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T10:27:05Z\\nNo. of bitstreams: 1\\nBertiProcesi05-1.pdf: 211758 bytes, checksum: b6c3ae059191cddb5c025aee61a23799 (MD5)