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Phase transition for invariant measures of the focusing Schrödinger equation

Leonardo Tolomeo, Hendrik Weber

TL;DR

This work rigorously analyzes Gibbs measures for the focusing nonlinear Schrödinger equation on the 1D torus in the large‑torus limit. By leveraging the Boué–Dupuis variational formula, the authors reduce the problem to a deterministic minimisation problem $A(\beta,\mathbf{N})$ and identify precise regime‑dependent behaviours as $L\to\infty$: (i) in the supercritical and strongly nonlinear critical cases the measure concentrates around a rescaled soliton and the partition function decays with a specific exponent, (ii) in the subcritical and weakly nonlinear critical cases the measure converges to the Ornstein–Uhlenbeck process, and (iii) at criticality a sharp phase transition occurs, including non‑trivial limits and conditions under which OU convergence fails. The results resolve long‑standing questions sparked by numerical conjectures and reconcile earlier negative results, providing a detailed picture of phase transitions in invariant Gibbs measures for focusing NLS. The methods unify variational techniques with probabilistic large‑deviation control to describe concentration phenomena in invariant measures and illuminate the interplay between nonlinearity, mass cutoff, and spatial scale. These findings have broad implications for the statistical mechanics of nonlinear dispersive waves and the rigorous understanding of Gibbs measures in low dimensions.

Abstract

We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus $\mathbb T$, that was introduced in a seminal paper by Lebowitz, Rose and Speer (1988). We show that in the large torus limit, the measure exhibits a phase transition, depending on the size of the nonlinearity. This phase transition was originally conjectured on the basis of numerical simulation by Lebowitz, Rose and Speer (1988). Its existence is however striking in view of a series of negative results by McKean (1995) and Rider (2002).

Phase transition for invariant measures of the focusing Schrödinger equation

TL;DR

This work rigorously analyzes Gibbs measures for the focusing nonlinear Schrödinger equation on the 1D torus in the large‑torus limit. By leveraging the Boué–Dupuis variational formula, the authors reduce the problem to a deterministic minimisation problem and identify precise regime‑dependent behaviours as : (i) in the supercritical and strongly nonlinear critical cases the measure concentrates around a rescaled soliton and the partition function decays with a specific exponent, (ii) in the subcritical and weakly nonlinear critical cases the measure converges to the Ornstein–Uhlenbeck process, and (iii) at criticality a sharp phase transition occurs, including non‑trivial limits and conditions under which OU convergence fails. The results resolve long‑standing questions sparked by numerical conjectures and reconcile earlier negative results, providing a detailed picture of phase transitions in invariant Gibbs measures for focusing NLS. The methods unify variational techniques with probabilistic large‑deviation control to describe concentration phenomena in invariant measures and illuminate the interplay between nonlinearity, mass cutoff, and spatial scale. These findings have broad implications for the statistical mechanics of nonlinear dispersive waves and the rigorous understanding of Gibbs measures in low dimensions.

Abstract

We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus , that was introduced in a seminal paper by Lebowitz, Rose and Speer (1988). We show that in the large torus limit, the measure exhibits a phase transition, depending on the size of the nonlinearity. This phase transition was originally conjectured on the basis of numerical simulation by Lebowitz, Rose and Speer (1988). Its existence is however striking in view of a series of negative results by McKean (1995) and Rider (2002).
Paper Structure (13 sections, 30 theorems, 298 equations)

This paper contains 13 sections, 30 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. Construction of the measure and variational formulation
  3. Supercritical case and strongly nonlinear critical case: concentration around a single soliton
  4. Weakly nonlinear critical case and subcritical case
  5. Structure of the paper
  6. Preliminaries
  7. Approximate process
  8. Analysis of the deterministic variational problem
  9. Construction of the measure
  10. Supercritical case and strongly nonlinear critical case
  11. Subcritical case and weakly nonlinear critical case
  12. The Buoé-Dupuis formula in infinite dimension
  13. Gaussian free field and Ornstein-Uhlenbeck measure

Key Result

Theorem 1.1

Let $2 < p < 6$, $\alpha, \mathbf{N} > 0$, and consider the measure $\rho_L$ formally given by Then the following statements hold. In all situations above, the convergence is to be understood as weak convergence in the space of probability measures over distributions over $\mathbb{R}$, where we see distributions over $L\mathbb{T} \cong [-L/2,L/2]$ as distributions over $\mathbb{R}$ via their per

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2: Lebowitz1988Bo94OST
  • Theorem 1.3: Supercritical case
  • Theorem 1.4: Critical case, strong nonlinearity
  • Corollary 1.5
  • Theorem 1.6: Weakly nonlinear critical case and subcritical case
  • Lemma 2.1
  • proof
  • Lemma 2.2: Gagliardo-Nirnenberg-Sobolev on the torus
  • proof
  • ...and 45 more