On the continuous resonant equation for NLS: II. Statistical study
Analysis of PDEs
2016-01-20 v1
Abstract
We consider the continuous resonant (CR) system of the 2D cubic nonlinear Schr{\"o}dinger (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.
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Cite
@article{arxiv.1502.05643,
title = {On the continuous resonant equation for NLS: II. Statistical study},
author = {Pierre Germain and Zaher Hani and Laurent Thomann},
journal= {arXiv preprint arXiv:1502.05643},
year = {2016}
}
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23 pages