70 research outputs found

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

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

We consider the cubic nonlinear Schr\"odinger equation posed on the spatial
domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and
construct modified wave operators for small initial and final data respectively
($1\leq d\leq 4)$. 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 $d\geq 2$. As a consequence, we obtain
global solutions to the defocusing and focusing problems on $\mathbb{R}\times
\mathbb{T}^d$ (for any $d\geq 2$) with infinitely growing high Sobolev norms
$H^s$.Comment: 47 pages. Minor corrections and several typos fixe

### Full derivation of the wave kinetic equation

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 $0$, according to a particular scaling law
(Boltzmann-Grad in the particle case).
More precisely, in dimensions $d\geq 3$, we consider the (NLS) equation in a
large box of size $L$ with a weak nonlinearity of strength $\alpha$. In the
limit $L\to\infty$ and $\alpha\to 0$, under the scaling law $\alpha\sim
L^{-1}$, 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 $O(1)$ (i.e independent of $L$ and $\alpha$) multiples of the
kinetic timescale $T_{\text{kin}}\sim \alpha^{-2}$. 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

- â€¦