Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 7 (2015), 1733-1756.

On the continuous resonant equation for NLS, II: Statistical study

Pierre Germain, Zaher Hani, and Laurent Thomann

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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.

Article information

Anal. PDE, Volume 8, Number 7 (2015), 1733-1756.

Received: 15 April 2015
Revised: 4 June 2015
Accepted: 29 July 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems

nonlinear Schrödinger equation random data Gibbs measure white noise measure weak solutions global solutions


Germain, Pierre; Hani, Zaher; Thomann, Laurent. On the continuous resonant equation for NLS, II: Statistical study. Anal. PDE 8 (2015), no. 7, 1733--1756. doi:10.2140/apde.2015.8.1733.

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