70 research outputs found

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

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    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.Comment: 23 page

    Modified scattering for the cubic Schr\"odinger equation on product spaces and applications

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    We consider the cubic nonlinear Schr\"odinger equation posed on the spatial domain R×Td\mathbb{R}\times \mathbb{T}^d. We prove modified scattering and construct modified wave operators for small initial and final data respectively (1d4)1\leq d\leq 4). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when d2d\geq 2. As a consequence, we obtain global solutions to the defocusing and focusing problems on R×Td\mathbb{R}\times \mathbb{T}^d (for any d2d\geq 2) with infinitely growing high Sobolev norms HsH^s.Comment: 47 pages. Minor corrections and several typos fixe

    Full derivation of the wave kinetic equation

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    We provide the rigorous derivation of the wave kinetic equation from the cubic nonlinear Schr\"odinger (NLS) equation at the kinetic timescale, under a particular scaling law that describes the limiting process. This solves a main conjecture in the theory of wave turbulence, i.e. the kinetic theory of nonlinear wave systems. Our result is the wave analog of Lanford's theorem on the derivation of the Boltzmann kinetic equation from particle systems, where in both cases one takes the thermodynamic limit as the size of the system diverges to infinity, and as the interaction strength of waves or radius of particles vanishes to 00, according to a particular scaling law (Boltzmann-Grad in the particle case). More precisely, in dimensions d3d\geq 3, we consider the (NLS) equation in a large box of size LL with a weak nonlinearity of strength α\alpha. In the limit LL\to\infty and α0\alpha\to 0, under the scaling law αL1\alpha\sim L^{-1}, we show that the long-time behavior of (NLS) is statistically described by the wave kinetic equation, with well justified approximation, up to times that are O(1)O(1) (i.e independent of LL and α\alpha) multiples of the kinetic timescale Tkinα2T_{\text{kin}}\sim \alpha^{-2}. This is the first result of its kind for any nonlinear dispersive system.Comment: 137 pages, 44 figures. [V3] Added a discussion of the recent work (arXiv:2106.09819) of Staffilani-Tra
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