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.

}, isbn = {2523-3688}, url = {https://doi.org/10.1007/s42286-020-00036-8}, author = {Massimiliano Berti and Roberto Feola and Luca Franzoi} } @article {Berti2013229, title = {Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential}, journal = {Journal of the European Mathematical Society}, volume = {15}, number = {1}, year = {2013}, note = {cited By (since 1996)5}, pages = {229-286}, abstract = {We prove the existence of quasi-periodic solutions for Schr{\"o}dinger 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$\infty$ then the solutions are C$\infty$. 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. {\textcopyright} European Mathematical Society 2013.}, issn = {14359855}, doi = {10.4171/JEMS/361}, author = {Massimiliano Berti and Philippe Bolle} } @article {2006, title = {Quasi-periodic solutions of completely resonant forced wave equations}, journal = {Comm. Partial Differential Equations 31 (2006) 959 - 985}, number = {SISSA;106/2004/M}, year = {2006}, abstract = {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.}, doi = {10.1080/03605300500358129}, url = {http://hdl.handle.net/1963/2234}, author = {Massimiliano Berti and Michela Procesi} } @article {2005, title = {Quasi-periodic oscillations for wave equations under periodic forcing}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116}, year = {2005}, publisher = {Accademia Nazionale dei Lincei}, url = {http://hdl.handle.net/1963/4583}, author = {Massimiliano Berti and Michela Procesi} }