A KAM theorem for the defocusing NLS equation

In this paper we prove a KAM theorem for the defocusing NLS equation in one space dimension with periodic boundary conditions. The novelty of our result is that it is valid not only near the zero solution, but on the entire Sobolev space HN(T,C) with N∈Z⩾1. In particular, the invariant tori which persist under small Hamiltonian perturbations might be far away from the zero potential.