Modified Scattering of Cubic Nonlinear Schr\"odinger Equation on Rescaled Waveguide Manifolds

TL;DR

This paper applies modified scattering theory to the cubic nonlinear Schrödinger equation on rescaled waveguide manifolds, showing small-data solutions have bounded Sobolev norms and exhibit weak instability.

Contribution

It extends modified scattering analysis to higher-dimensional waveguide manifolds, revealing boundedness and instability properties of solutions.

Findings

Sobolev norms remain bounded for small-data solutions

Solutions exhibit weak instability

Analysis applies to manifolds with dimension d ≥ 2

Abstract

We use modified scattering theory to demonstrate that small-data solutions to the cubic nonlinear Schr\"odinger equation on rescaled waveguide manifolds, $\mathbb{R} \times \mathbb{T}^d$ for $d\geq 2$, demonstrate boundedness of Sobolev norms as well as weak instability.