Phase Transition for Invariant Measures of the Focusing Schrödinger Equation

Abstract We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus $${\mathbb {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> , 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).