On the continuous resonant equation for NLS II: Statistical study

Pierre Germain, Zaher Hani, Laurent Thomann

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1733-1756
Number of pages24
JournalAnalysis and PDE
Volume8
Issue number7
DOIs
StatePublished - 2015

Fingerprint

White noise
Sobolev spaces
Invariance
Nonlinear equations
Cubic equation
Trapping
Invariant Measure
Global Solution
Sobolev Spaces
Nonlinear Equations

Keywords

  • Gibbs measure
  • Global solutions
  • Nonlinear Schrödinger equation
  • Random data
  • Weak solutions
  • White noise measure

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Numerical Analysis

Cite this

On the continuous resonant equation for NLS II : Statistical study. / Germain, Pierre; Hani, Zaher; Thomann, Laurent.

In: Analysis and PDE, Vol. 8, No. 7, 2015, p. 1733-1756.

Research output: Contribution to journalArticle

Germain, Pierre ; Hani, Zaher ; Thomann, Laurent. / On the continuous resonant equation for NLS II : Statistical study. In: Analysis and PDE. 2015 ; Vol. 8, No. 7. pp. 1733-1756.
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