Discrete Bound States in a Toy Model for Weak Turbulence and Implications for the Invariant Measure

TL;DR

This paper introduces a simplified model to analyze weak turbulence in nonlinear Schrödinger equations, focusing on energy minimizers and invariant measures to understand frequency cascades.

Contribution

It develops a framework for classifying energy minimizers and characterizing invariant measures in a toy model for weak turbulence.

Findings

Identification of energy minimizers for fixed mass

Characterization of invariant measures near minimizers

Insights into frequency cascade mechanisms

Abstract

A model Hamiltonian dynamical system has been derived to study frequency cascades in the cubic defocusing nonlinear Schr\"odinger equation on the torus. Here, we explore the framework for exploring a canonical ensemble formulation of the dynamics through classification of energy minimizers for fixed mass and characterizing the invariant measure in a neighborhood of those minimizers.