Stability results for the nonlinear Schrödinger equation on Diophantine tori

We present two results of enhanced long-time stability for the nonlinear Schrödinger equation posed on rescaled tori with Diophantine properties. This proceeding for the conference “Journées Équations aux dérivées partielles” is the opportunity to give a glimpse on modern methods at the interface of PDEs and classical mechanics that are currently being developed to study the long-time dynamics of Hamiltonian resonant PDEs.