a) KAM for water waves
b) Birkhoff normal form for water waves
c) KAM for 2d Euler and quasi linear KdV
d) Modulational instability of Stokes waves
e) KAM for NLS and NLW
f) Arnold diffusion and chaotic dynamics
g) Zoll magnetic surfaces
Below a selection of papers with a short description of the major results
a) KAM for water waves. The water waves equations, written by Laplace and Lagrange in the 18 century, are a particularly challenging free boundary problem, governed by an infinite dimensional Hamiltonian system with a quasi-linear and non-local vector field. The next works contain the first existence results of time quasi-periodic solutions, either standing and traveling, of the water waves equations:
P. Baldi, M. Berti, E. Haus, R. Montalto, “Time quasi-periodic gravity water waves in finite depth”, Inventiones Mathematicae, 214, 739–911, 2018, 10.1007/s00222-018-0812-2
M. Berti, R. Montalto, “KAM for gravity capillary water waves”, Memoires AMS, Volume 263, 1273, ISSN 0065-9266, 2020.
M. Berti, L. Franzoi, A. Maspero, “Pure gravity traveling quasi-periodic water waves with constant vorticity”, Comm. Pure Applied Mathematics, 77, 2, 990-1064, 2024,.10.1002/cpa.22143
M. Berti, L. Franzoi, A. Maspero, “Traveling quasi-periodic water waves with constant vorticity”, Arch. Rat. Mech., 240, 99-202, 2021. 10.1007/s00205-021-01607
A key idea was the introduction of pseudo-differential calculus and Fourier integral operators for the spectral analysis of the linearized operators along a Nash-Moser implicit function iteration
b) Birkhoff normal form for water waves. The next results are the first and more general long time existence results of space periodic water waves.
M. Berti, J-M. Delort, “Almost global solutions for capillarity-gravity water waves equations on the circle”, UMI Lecture Notes, 24, x+268, 2018. Monograph awarded for the UMI book prize. ISBN 978-3-319-99485-7
M. Berti, R. Feola, F. Pusateri, “Birkhoff normal form and long time existence for periodic gravity water waves”, Comm. Pure Applied Mathematics, doi/10.1002/cpa.22041, 76, 7, 1416-1494, 2023
M. Berti, A. Maspero, F. Murgante, ``Hamiltonian Birkhoff normal form for gravity-capillary water waves with constant vorticity: almost global existence", Annals of PDEs, 10, 22, 2024. https://doi.org/10.1007/s40818- 024-00182-z
Key ideas are the introduction of a para-differential normal form –a non-linear analogue of pseudo-differential reduction in decreasing order– and the implementation of a novel reversible/Hamiltonian Birkhoff normal form for quasi-linear PDEs.
A recent application to also SQG-alpha equations has been proved here:
M. Berti, S. Cuccagna, F. Gancedo, S. Scrobogna, Paralinearization and extended lifespan for solutions of the α-SQG sharp front equation, Advances in Math. 460, 2025
c) KAM for 2d Euler and quasi linear KdV. The next work contains the first existence results of time quasi-periodic vortex patch solutions of the 2d-Euler equations, showing a very rich dynamics close to the Kirkhoff ellipses.
M. Berti, Z. Hassaina, N. Masmoudi, "Time quasi-periodic vortex patches", Inventiones Mathematicae, 233, 1279-1391, link.springer.com/article/10.1007/s00222-023-01195-4
The next works prove the stability of multi- periodic solitons of KdV, either small and large, under quasi-linear perturbations.
M. Berti, T. Kappeler, R. Montalto, “Large KAM tori for quasi-linear perturbations of KdV”, Arch. Rat. Mech., 239, 1395-1500, 10.1007/s00205-020-01596-2, 2021
P. Baldi, M. Berti, R. Montalto, “KAM for autonomous quasi-linear perturbations of KdV”, Annales I. Poincaré, 33, 1589-1638, 2016
P. Baldi, M. Berti, R. Montalto “KAM theory for quasi-linear and fully nonlinear forced perturbations of Airy equations”, Math. Annalen, 359, 1, 471-536, 2014
Among several new tools these works introduced the idea of “weak” and ‘linear” Birkhoff normal forms, combined with integrable systems methods, to obtain non-resonance conditions among the frequencies tuning the initial data, for quasi-linear PDEs.
d) Modulational instability of Stokes waves. A problem of fundamental importance in fluid mechanics regards the stability/instability of traveling periodic Stokes waves, i.e. traveling solutions of the pure gravity water waves. The next results fully describe the unstable spectral bands of small amplitude Stokes waves.
M. Berti, A. Maspero, P. Ventura, “Full description of Benjamin-Feir instability of Stokes waves in deep water”, Inventiones Mathematicae, 10.1007/s00222-022-01130-z doi.org/10.1007/s00222-022-01130-z, 230, 2, 651--711, 2022.
M. Berti, A. Maspero, P. Ventura, “Benjamin-Feir instability of Stokes waves in finite depth”, Arch. Rational Mech., 247, 91, 2023, 10.1007/s00205-023-01916-2
M. Berti, A. Maspero, P. Ventura, ``Stokes waves at the critical depth are modulational unstable", Comm. Math. Phys., 405-56, 2024, https://doi.org/10.1007/s00220-023-04928-x
M. Berti, L. Corsi, A. Maspero, P. Ventura, Infinitely many isolas of modulational instability for Stokes waves, arxiv.org/abs/2405.05854, 2024.
The proofs develop a novel symplectic version of Kato’s theory of similarity transformations combined with a KAM inspired block diagonalization. This approach has yet become a classical tool in the field (e.g. used in several new papers by W. Strauss and collaborators).
e) KAM for NLS and NLW. These works prove existence of periodic and quasi-periodic solutions of Nonlinear wave and Schrodinger equations in completely resonant cases and in higher space dimension.
M. Berti, P. Bolle, “Quasi-periodic solutions with Sobolev regularity of NLS on T^d and a multiplicative potential”, JEMS, 15, 229-286, 2013 DOI 10.4171/JEMS/361
M. Berti, P. Bolle, “Quasi-periodic solutions of nonlinear wave equations on T^d”, vii + 355, Monographs of the EMS, doi 10.4171/211 ISBN print 978-3-03719-211-5, 2020
M. Berti, M.Procesi, “Nonlinear wave and Schrodinger equations on compact Lie groups and homogeneous spaces”, Duke Math. Jour., 159, 3, 479-538, 2011, DOI: 10.1215/00127094-1433403
M. Berti, P. Bolle: “Periodic solutions of nonlinear wave equations with general nonlinearities”, Comm. in Math. Phys., Vol. 243, 2, 315-328, 2003
M. Berti, P. Bolle, “Cantor families of periodic solutions for completely resonant nonlinear wave equations”, Duke Math. Jour., 134, 2, 359-419, 2006, DOI: 10.1215/S0012-7094-06-13424-5
M. Berti, L. Biasco, M. Procesi, “KAM theory for the Hamiltonian derivative wave equation”, Annales Scientifiques de l’ ENS, 46, 2, 299-371, 2013.
M. Berti, B. Langella, D. Silimbani, ``Time periodic solutions of completely resonant Klein-Gordon equations on S^3", Annales de l'I. Poincaré, DOI 10.4171/AIHPC/125, 2024
Some of the proofs extend the multiscale analysis (originally due to Bourgain) for the inversion of linearized operators in presence of huge blocks of small divisors which arise for the high degeneracy of the eigenvalues of the Laplacian in higher space dimension. For dealing with bifurcation of periodic solutions of completely resonant wave equations -they possesses an infinite dimensional Kernel- we use critical point theory combined with Strichartz estimates for the linear flow.
f) Arnold diffusion a chaotic dynamics. The next two papers proved the existence of unstable trajectories of finite dimensional nearly integrable Hamiltonian systems, which drift with optimal diffusion times:
M. Berti, L. Biasco, P. Bolle: “Drift in phase space: a new variational mechanism with optimal diffusion time”, Journal des Math. Pures et Appliquées, https://doi.org/10.1016/S0021-7824(03)00032-1 82/6, pp. 613-664, 2003
M. Berti, P. Bolle: “A functional analysis approach to Arnold Diffusion”, Annales de l'l. Poincaré, DOI 10.1016/S0294-1449(01)00084-1 19, 4, 395-450, 2002
The techniques of proof are based on variational methods in bifurcation theory to construct multi-bump chaotic trajectories, as well for the next two papers.
M. Berti, P. Bolle: “Variational construction of Homoclinics and Chaotic Behaviour in presence of a saddle-saddle equilibrium”, Annali Scuola Normale Superiore di Pisa, IV, XXVII, 2, 1998.
M. Berti, A. Malchiodi: “Non-compactenss and multiplicity results for the Yamabe problem on Sn”, Journal of Functional Analysis, 180, 1, 2001.
The latter work proves the existence of metrics on the sphere S^n close to standard one for which the Yamabe problem possesses a sequence of unbounded solutions. It is inserted here as the construction is inspired by multibump chaotic dynamics.
g) Zoll magnetic surfaces. This very recent paper proves that the space of Zoll magnetic flows on the two-torus -i.e. metric and magnetic systems with surface completely filled by periodic orbits– is infinite dimensional :
L. Asselle, G. Benedetti, M. Berti, “Zoll magnetic systems on the two-torus: a Nash-Moser construction”, Advances in Math., 452, 2024
The proof is based on a Nash-Moser implicit function theorem with new arguments involving techniques from microlocal analysis. This a very promising new area of research: in the Riemannian setting there was only a previous result on S2 by Guillemin in ’75