Energy transfer model for the derivative nonlinear Schrodinger equations on the torus

TL;DR

This paper studies energy transfer in a derivative nonlinear Schrödinger equation on a torus, revealing how resonant interactions influence Sobolev norm growth and energy distribution among modes.

Contribution

It identifies the role of resonant clusters and phase matching in energy transfer dynamics for the derivative NLS on the torus.

Findings

Energy transfer is governed by four initially excited frequency modes.

Resonant interactions lead to growth in Sobolev norms.

Energy remains confined within certain frequency clusters.

Abstract

We consider the nonlinear derivative Schrodinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties consist of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. This phenomena arrests energy transfers between low and high modes, which are quantified by a growth in the Sobolev norm.