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On the continuous
resonant equation for NLS, II: Statistical study
Pierre Germain, Zaher Hani and Laurent Thomann |
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Vol. 8 (2015), No. 7, 1733–1756
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35Q55, 37K05, 37L50
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Milestones
Received: 15 April 2015
Revised: 4 June 2015
Accepted: 29 July 2015
Published: 18 September 2015
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Authors |
| Pierre Germain | |
| Courant Institute of Mathematical
Sciences New York University 251 Mercer Street New York, NY 10012-1185 United States |
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| Zaher Hani | |
| School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 United States |
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| Laurent Thomann | |
| Laboratoire de Mathématiques J.
Leray Université de Nantes UMR 6629 du CNRS 2, rue de la Houssinière 44322 Nantes 03 France |
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