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<p>We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {T}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation>, that was introduced in a seminal paper by Lebowitz et al. (J Stat Phys 50(3):657—687, 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 et al. (J Stat Phys 50(3):657—687, 1988). Its existence is however striking in view of a series of negative results by McKean (Commun Math Phys 168(3):479—491, 1995) and Rider (Commun Pure Appl Math 55(10):1231—1248, 2002).</p>

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Phase Transition for Invariant Measures of the Focusing Schrödinger Equation

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  • 发表日期：

    2026年3月24日

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  • Leonardo Tolomeo,
  • Hendrik Weber

摘要

We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus \({\mathbb {T}}\) T , that was introduced in a seminal paper by Lebowitz et al. (J Stat Phys 50(3):657—687, 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 et al. (J Stat Phys 50(3):657—687, 1988). Its existence is however striking in view of a series of negative results by McKean (Commun Math Phys 168(3):479—491, 1995) and Rider (Commun Pure Appl Math 55(10):1231—1248, 2002).