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All the almost periodic solutions for nonintegrable PDEs found in the literature are very regular (at least ) 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.
We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sine-Gordon kink, a remarkable nonresonance property exhibited by the quadratic nonlinearity in the Klein–Gordon equation for the perturbation, and a variable coefficient quadratic normal form. We emphasize that the restriction to odd perturbations does not bypass the effects of the odd threshold resonance of the linearized operator. Our techniques have applications to soliton stability questions for several well-known nonintegrable models, for instance, to the asymptotic stability problem for the kink of the model as well as to the conditional asymptotic stability problem for the solitons of the focusing quadratic and cubic Klein–Gordon equations in one space dimension.