Modified scattering for the cubic Schr\"odinger equation on Diophantine waveguides

TL;DR

This paper studies the long-term behavior of small solutions to the cubic Schrödinger equation on waveguides satisfying Diophantine conditions, showing they undergo modified scattering without Sobolev norm growth, contrasting with non-Diophantine cases.

Contribution

It demonstrates modified scattering for the cubic Schrödinger equation on Diophantine waveguides, revealing stability and boundedness of Sobolev norms unlike previously observed energy cascades.

Findings

Solutions exhibit modified scattering to an effective dynamics.

Sobolev norms remain bounded over time.

Contrasts with non-Diophantine scenarios showing energy cascades.

Abstract

We consider the cubic Schr\"odinger equation posed on product spaces subject to a generic Diophantine condition. Our analysis shows that the small-amplitude solutions undergo modified scattering to an effective dynamics governed by some interactions that do not amplify the Sobolev norms. This is in sharp contrast with the infinite energy cascade scenario observed by Hani--Pausader--Tzvetkov--Visciglia in the absence of Diophantine conditions.