Small amplitude weak almost periodic solutions for the one-dimensional NLS

All the almost periodic solutions for non integrable PDEs found in the literature are very regular (at least C1) and, hence, very close to quasi periodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with  nite regularity is a main open problem in KAM theory for PDEs. Here we consider the one dimensional NLS with external parameters and construct almost periodic solutions which have only Sobolev regularity both in time and space. Moreover many of our solutions are so only in a weak sense. This is the  rst result on existence of weak, i.e. non classical, solutions for non integrable PDEs in KAM theory.