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

Abstract

We revisit the question of an invariant measure for the focusing cubic Schrö-dinger 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.

Probabilistic Methods for Discrete Nonlinear Schrödinger Equations

“…The last two displays and the inequalities (33) and (34) Since q > 17/18, it follows that < 1/7 and therefore 1 − 2 /5 > 4/5 + , a fact that will be needed below. If M 1 (ψ) > n 1−2 /5 , then S 4 (ψ) > n 2−4 /5 and therefore ψ ∈ Γ a,b for some n −2 /5 < a < b ≤ B.…”

“…Studying the nature of the invariant measures may yield important information about the long-term behavior of these systems. The only results we know in this direction are those of Brydges and Slade [7] in d = 2 and Rider [33,34] in d = 1. Some progress for invariant measures of the KdV equation has been made recently in [30].…”

Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

“…In [11], Bourgain wrote "It seems worthwhile to investigate this aspect [the (non-)normalizability issue of the focusing Gibbs measures] more as a continuation of [51] and [16] ." See related works [12,19,72,73,78,86,90] on the non-normalizability (and other issues) for focusing Gibbs measures.…”

Invariance of white noise for KdV on the line

“…We turn now to the question of taking the infinite-volume limit of these measures. This has been investigated by Rider [49,50], who considers the thermodynamic limit of NLS in one dimension, and by Chatterjee [9], who considers a simultaneous continuum and infinite-volume limit of discretized NLS in general dimension at fixed mass and energy. In both cases, the limit was proven not to exist.…”