Energy cascades for NLS on the torus

TL;DR

This paper investigates energy transfer in the nonlinear Schrödinger equation on a torus, demonstrating how small initial data can lead to large mode creation through a specialized analytical approach.

Contribution

It introduces a novel analysis of energy cascades for NLS on the torus using multiphase weakly nonlinear geometric optics and a discrete dynamical system.

Findings

Small initial data with five modes can generate arbitrarily large modes at specific times.

The proof employs reduction to a two-dimensional discrete dynamical system.

Energy cascades occur in the studied NLS setting, revealing complex mode interactions.

Abstract

We consider the nonlinear Schrodinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.