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All the almost periodic solutions for non integrable PDEs found in theliterature are very regular (at least $C^\infty$) and, hence, very close toquasi periodic ones. This fact is deeply exploited in the existing proofs.Proving the existence of almost periodic solutions with finite regularity is amain open problem in KAM theory for PDEs. Here we consider the one dimensionalNLS with external parameters and construct almost periodic solutions which haveonly Sobolev regularity both in time and space. Moreover many of our solutionsare so only in a weak sense. This is the first result on existence of weak,i.e. non classical, solutions for non integrable PDEs in KAM theory.

Small amplitude weak Sobolev almost periodic solutions for the 1d NLS