In this talk, I will talk about the quasi-periodic solutions of Hamiltonian perturbations of higher dimensional NLS and NLW equations.
One of the approaches to constructing quasi-periodic solutions is by KAM in infinitely dimensional phases. I listed some authors here Bambusi, Eliasson, Grébert, Kuksin, Wayne, Poschel, Procesi, .......
Another approach was initiated by Craig and Wayne based on Lyapunov-Schmidt decomposition, which splits the problem into a Q-equation and P-equation. This approach was highly developed by Bourgain. This approach essentially is the control of inverse of certain linear operators, which arise from consecutive linearizations when applying the Newton iteration scheme. It turns out that the control of inverse of the matrices is closely related to the proof of Anderson localization of quasi-periodic operators. With the additional help of separate properties of eigenvalues (arithmetic lemma), Bourgain constructed the quasi-periodic of higher dimensional NLS and NLW equations.
I am going to use two lectures to talk about Bourgain's proof. I will talk about the small divisor problems, the large deviation theorem, semi-algebraic sets and the arithmetic lemma.