Small amplitude weak almost periodic solutions for the 1d NLS

All the almost periodic solutions for nonintegrable PDEs found in the literature are very regular (at least C-infinity) and, hence, very close to quasiperiodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with finite regularity is a main open problem in KAM theory for PDEs. Here we consider the 1-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 first result on existence of weak (i.e., nonclassical) solutions for nonintegrable PDEs in KAM theory.