Almost sure scattering for the defocusing cubic nonlinear Schrödinger equation on [formula omitted]

Abstract

We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation (NLS) on the waveguide manifold R3×T and establish almost sure scattering for random initial data, where no symmetry conditions are imposed and the result is available for arbitrarily rough data f∈Hs with s∈R. The main new ingredient is a layer-by-layer refinement of the newly established randomization introduced by Shen-Soffer-Wu [40], which enables us to also obtain a strongly smoothing effect from the randomization for the forcing term along the periodic direction. It is worth noting that such a smoothing effect generally can not hold for purely compact manifolds, which is on the contrary available for the present model thanks to the mixed type nature of the underlying domain. As a byproduct, by assuming that the initial data are periodically trivial, we also obtain almost sure scattering for the defocusing cubic NLS on R3 which parallels the result by Camps [15] and Shen-Soffer-Wu [41]. To our knowledge, the paper also gives the first almost sure well-posedness and scattering result for NLS on product spaces.