Some new results on almost periodic solutions for dispersive PDEs on the circle

Existence of almost periodic solutions for evolution PDEs is a very interesting problem, with a lot of open questions. Most of the literature is on the construction of very regular solutions for semilinear PDEs (mainly the NLS) with external parameters. I shall discuss two new results: 1. existence of solutions for a quasi-linear forced Airy equation, 2. existence of finite regularity solutions for the traslation invariant NLS equation.