Beating effects in cubic Schr\"odinger systems and growth of Sobolev norms

TL;DR

This paper demonstrates energy exchange phenomena in coupled cubic Schrödinger systems and constructs solutions with logarithmic Sobolev norm growth, highlighting dynamic behaviors over finite times.

Contribution

It introduces the existence of beating effects and Sobolev norm growth in coupled cubic Schrödinger systems, extending to linear equations.

Findings

Existence of beating effects in coupled systems

Construction of solutions with logarithmic Sobolev norm growth

Results valid for large but finite times

Abstract

We consider a system of coupled cubic Schr\"odinger equations. We prove that there exists a beating effect, i.e. an energy exchange between different modes. This construction may be transported to the linear time-dependent Schr\"odinger equation: we build solutions such that their Sobolev norms grow logarithmically. All these results are stated for large but finite times.