|University of Basel
Quasi-periodic property is a commonly observed property in many Hamiltonian systems. While such a property is easily observed in a linear Hamiltonian system, it is much more complicated to prove whether such solutions can exist in a nonlinear Hamiltonian system. The KAM theory is a classical method used to construct quasi-periodic solutions in a nonlinear/perturbed system. In this lecture, I will outline a proof of an application of the KAM theory to the generalized-surface quasi-geostrophic equations, constructing quasi-periodic solutions near a Rankine vortex.
Lecture notes on nonlinear oscillations of Hamiltonian PDEs (Chapter: A tutorial in Nash-Moser theory) by M. Berti
KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation by P. baldi, M. Berti and R. Montalto
KAM for autonomous quasi-linear perturbations of KdV by P. baldi, M. Berti and R. Montalto.
Quasiperiodic solutions of the generalized SQG equation by J. Gomez-Serrano, A. Ionescu, J. Park