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

Jeremy L. Marzuola, Jonathan C. Mattingly

TL;DR

The paper studies a discrete toy model for resonant interactions in the defocusing cubic NLS on a torus and develops a canonical Gibbs framework on fixed-mass shells. It proves that the global energy minimizer at fixed mass is a unique three-mode in-phase state with explicit amplitudes, and analyzes the local geometry via the Lagrange multiplier and Hessian. In the low-temperature limit, the invariant measure concentrates near the minimizer orbit, with leading-order fluctuations described by a Gaussian measure supported on a five-mode neighborhood, while regions away from this neighborhood contribute negligibly as $\beta\to\infty$. The results provide a precise probabilistic description of the measure near minimizers, revealing a structured, localized fluctuation pattern that informs our understanding of weak turbulence and invariant measures in discrete and continuum NLS settings.

Abstract

A model Hamiltonian dynamical system has been derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger 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.

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

TL;DR

The paper studies a discrete toy model for resonant interactions in the defocusing cubic NLS on a torus and develops a canonical Gibbs framework on fixed-mass shells. It proves that the global energy minimizer at fixed mass is a unique three-mode in-phase state with explicit amplitudes, and analyzes the local geometry via the Lagrange multiplier and Hessian. In the low-temperature limit, the invariant measure concentrates near the minimizer orbit, with leading-order fluctuations described by a Gaussian measure supported on a five-mode neighborhood, while regions away from this neighborhood contribute negligibly as . The results provide a precise probabilistic description of the measure near minimizers, revealing a structured, localized fluctuation pattern that informs our understanding of weak turbulence and invariant measures in discrete and continuum NLS settings.

Abstract

A model Hamiltonian dynamical system has been derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger 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.
Paper Structure (21 sections, 25 theorems, 150 equations, 3 figures)

This paper contains 21 sections, 25 theorems, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Toy Model
  3. Gibbs Measures
  4. Low temperature behavior
  5. Connections to Previous Work and outline of the result
  6. Organization of Paper
  7. Exact in phase solutions
  8. The 3 mode solution
  9. Energy Minimizers for Fixed Mass
  10. Linearizing about the 3-mode Minimizers
  11. The Lagrange Multiplier
  12. The Hessian Calculation
  13. A Gaussian approximation
  14. Properties the Gaussian Measures
  15. Defining the measure with intrinsic coordinates on the sphere
  16. ...and 6 more sections

Key Result

Theorem 1

For a given mass $m>0$, there exists a 3-node solution $\mathbf{b}^*$ that is a minimizer of $H$ with $M(\mathbf{b}^*)=m$. It is the unique minimizer of $H$ up to translations and phase rotations. Namely for each lattice site $k \in \{-N+1,\dots,N-1\}$ and phase $\theta \in [0,2\pi)$, there exists a

Figures (3)

  • Figure 1: Solutions of \ref{['e:upmat']} with $\omega =1$. The solution at $N=5$ is not strictly positive.
  • Figure 2: Solutions of \ref{['e:upmat']} with $\omega =1$. Both solutions are strictly positive.
  • Figure 3: A cartoon of the coordinate system on the sphere with the representative spherical cap shaded and bordered in blue.

Theorems & Definitions (51)

  • Theorem 1
  • Remark 1.1
  • Proposition 1.1
  • proof
  • Remark 1.2
  • Corollary 1.2
  • Remark 1.3
  • Proposition 1.3
  • Theorem 2
  • Remark 1.4
  • ...and 41 more