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

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.