Behavior of a Model Dynamical System with Applications to Weak Turbulence

TL;DR

This paper experimentally investigates a model Hamiltonian dynamical system related to weak turbulence in the nonlinear Schrödinger equation, revealing energy cascades, stationary and periodic solutions, and hyperbolic behavior.

Contribution

It provides the first experimental analysis of frequency cascades and stationary solutions in a model system for weak turbulence, using a hydrodynamic reformulation.

Findings

Energy cascades are shown to be a generic phenomenon.

Identification of stationary and periodic solutions.

Evidence of hyperbolic behavior in the system.

Abstract

We experimentally explore solutions to a model Hamiltonian dynamical system derived in Colliander et al., 2012, to study frequency cascades in the cubic defocusing nonlinear Schr\"odinger equation on the torus. Our results include a statistical analysis of the evolution of data with localized amplitudes and random phases, which supports the conjecture that energy cascades are a generic phenomenon. We also identify stationary solutions, periodic solutions in an associated problem and find experimental evidence of hyperbolic behavior. Many of our results rely upon reframing the dynamical system using a hydrodynamic formulation.