Infinite dimensional invariant tori for nonlinear Schrödinger equations

We prove that nonlinear Schrödinger equations on the circle, without external parameters, admits plenty of almost periodic solutions. Indeed, we prove that arbitrarily close to most of the finite dimensional KAM tori constructed by Kuksin–Poschel in 1996, there exist infinite dimensional non resonant Kronecker tori, i.e. rotational invariant tori. This result answers a natural and longstanding question, well identified by the Hamiltonian PDE community since the first KAM-type result for PDEs by Kuksin in 1987.