On the ∞‐volume limit of the focusing cubic Schrödinger equation

Source abstract

Abstract We revisit the question of an invariant measure for the focusing cubic Schrödinger equation on the line. For the periodic problem the appropriate ensemble was introduced by Lebowitz, Rose, and Speer [3] and proved to be invariant under the flow by McKean [5]. These parties and others have also discussed the thermodynamic limit, though without consensus. Simulations carried out in [3] indicated the possibility of a phase transition. Similar experiments in [1] appeared to contradict that interpretation. Later, a proof was put forward in [6] that the full thermodynamic limit did not exist, suggesting a possible explanation for the disparate conclusions drawn from the numerics. Unfortunately, the latter contains an error. The main result here is that, in the infinite volume, the ensemble collapses onto the unit mass on the trivial path. Along the way sharp estimates for the partition function are established. © 2002 Wiley Periodicals, Inc.