Cancellations of Resonances and Long Time Dynamics of Cubic Schr\"odinger Equation on $\mathbb{T}$

TL;DR

This paper demonstrates the absence of wave turbulence in the 1D cubic nonlinear Schrödinger equation on a periodic domain by analyzing resonance cancellations and long-time dynamics using normal form transformations.

Contribution

It introduces a novel vanishing property of the normal form transformation and applies it to describe long-time behavior, preventing wave turbulence in 1D cubic NLS.

Findings

Normal form transformation vanishes for certain resonances

Long-time dynamics up to T=L^2/ε^4 are characterized

Wave turbulence behavior is shown to be absent in this setting

Abstract

We prove a vanishing property of the normal form transformation of the 1D cubic nonlinear Schr\"odinger (NLS) equation with periodic boundary conditions on $[0,L]$. We apply this property to quintic resonance interactions and obtain a description of dynamics for time up to $T=\frac{L^2}{\epsilon^4}$, if $L$ is sufficiently large and size of initial data $\epsilon$ is small enough. Since $T$ is the characteristic time of wave turbulence, this result implies the absence of wave turbulence behavior of 1D cubic NLS. Our approach can be adapted to other integrable systems without too many difficulties. In the proof, we develop a correspondence between Feynman diagrams and terms in normal forms, which allows us to calculate the coefficients inductively.