Vol. 8, No. 7, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
On the continuous resonant equation for NLS, II: Statistical study

Pierre Germain, Zaher Hani and Laurent Thomann

Vol. 8 (2015), No. 7, 1733–1756
Abstract

We consider the continuous resonant (CR) system of the 2-dimensional cubic nonlinear Schrödinger (NLS) equation. This system arises in numerous instances as an effective equation for the long-time dynamics of NLS in confined regimes (e.g., on a compact domain or with a trapping potential). The system was derived and studied from a deterministic viewpoint in several earlier works, which uncovered many of its striking properties. This manuscript is devoted to a probabilistic study of this system. Most notably, we construct global solutions in negative Sobolev spaces, which leave Gibbs and white noise measures invariant. Invariance of white noise measure seems particularly interesting in view of the absence of similar results for NLS.

Keywords
nonlinear Schrödinger equation, random data, Gibbs measure, white noise measure, weak solutions, global solutions
Mathematical Subject Classification 2010
Primary: 35Q55, 37K05, 37L50
Milestones
Received: 15 April 2015
Revised: 4 June 2015
Accepted: 29 July 2015
Published: 18 September 2015
Authors
Pierre Germain
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012-1185
United States
Zaher Hani
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332
United States
Laurent Thomann
Laboratoire de Mathématiques J. Leray
Université de Nantes
UMR 6629 du CNRS
2, rue de la Houssinière
44322 Nantes 03
France