Stability of the Cubic Nonlinear Schrodinger Equation on Irrational Tori

TL;DR

This paper investigates how the geometry of the domain, specifically irrational tori, affects the energy transfer phenomena in the defocusing cubic nonlinear Schrödinger equation, showing that irrational tori hinder the transfer of energy to high modes.

Contribution

The study demonstrates that on irrational tori, the energy transfer to high modes in the NLSE is less detectable compared to rational tori, highlighting the influence of domain geometry on solution dynamics.

Findings

Energy transfer is more difficult to detect on irrational tori.

The geometry of the domain influences the transfer of energy to high modes.

Differences between rational and irrational tori impact solution behavior in NLSE.

Abstract

A characteristic of the defocusing cubic nonlinear Schr\"odinger equation (NLSE), when defined so that the space variable is the multi-dimensional square (hence rational) torus, is that there exist solutions that start with arbitrarily small norms Sobolev norms and evolve to develop arbitrarily large modes at later times; this phenomenon is recognized as a weak energy transfer to high modes for the NLSE. In this paper, we show that when the system is considered on an irrational torus, energy transfer is more difficult to detect.