Large data global well-posedness and scattering for the focusing cubic nonlinear Schrödinger equation on $\mathbb{R}^2\times\mathbb{T}$

Abstract

We consider the focusing cubic nonlinear Schrödinger equation (NLS)Different from the 3D Euclidean case, the (3NLS) is mass-critical and non-scale-invariant on the waveguide manifold R 2 × T, hence the underlying analysis becomes more subtle and challenging. We formulate thresholds using the 2D Euclidean ground state of the focusing cubic NLS and show that solutions of (3NLS) lying below the thresholds are global and scattering in time. The proof relies on several new established Gagliardo-Nirenberg inequalities, whose best constants are formulated in term of the 2D Euclidean ground state. It is also worth noting the interesting fact that the thresholds for global well-posedness and scattering do not coincide. To the author's knowledge, this paper also gives the first large data scattering result for focusing NLS on product spaces.

On scattering for generalized NLS on waveguide manifolds

“…Threshold assumptions are necessary and new ingredients are needed to handle this type of problems. See [45] for a recent global well-posedness result, see [12,13,25,27] for the Euclidean results and see [6,31] for some very recent scattering result. 4.…”

On scattering asymptotics for the 2D cubic resonant system

“…Threshold assumptions are necessary and new ingredients are needed to handle this type of problems. See [37] for a global well-posedness result and [7,28] for two very recent scattering result. Moreover, see [14,24,27] for the Euclidean result.…”