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We consider a class of $1D$ NLS perturbed with a steplike potential. We provethat the nonlinear solutions satisfy the double scattering channels in theenergy space. The proof is based on concentration-compactness/rigidity method.We prove moreover that in dimension higher than one, classical scattering holdsif the potential is periodic in all but one dimension and is steplike andrepulsive in the remaining one.

Double Scattering Channels for $1D$ NLS in the Energy Space and its Generalization to Higher Dimensions