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

TL;DR

This paper establishes double scattering channels for 1D nonlinear Schrödinger equations with steplike potentials and extends classical scattering results to higher dimensions under specific potential conditions.

Contribution

It introduces the concept of double scattering channels in 1D NLS with steplike potentials and generalizes classical scattering results to higher dimensions with periodic and steplike potentials.

Findings

Proves double scattering channels in 1D energy space.

Shows classical scattering in higher dimensions with specific potential structures.

Utilizes concentration-compactness/rigidity method for proofs.

Abstract

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