Searcharxiv⌕ Search

arXiv subjects

N. Visciglia

Publications and source records attributed to N. Visciglia.

10 recordsLinked to original sources
arXiv2022

Global dynamics of the $2d$ NLS with white noise potential and generic polynomial nonlinearity

Using an approach introduced by Hairer-Labb\' e we construct a unique global dynamics for the NLS on $\T^2$ with a white noise potential and an arbitrary polynomial nonlinearity. We build the solutions as a limit of classical solutions (up to a phase shift) of the same equation with smoothed potentials. This is an improvement on previous contributions of us and Debussche-Weber dealing with quartic nonlinearities and cubic nonlinearities respectively.

math.PR↗
arXiv2021

$H^1$ scattering for mass-subcritical NLS with short-range nonlinearity and initial data in $Σ$

We consider short-range mass-subcritical nonlinear Schrödinger equations and we show that the corresponding solutions with initial data in $Σ$ scatter in $H^1$. Hence we up-grade the classical scattering result proved by Yajima and Tsutsumifrom $L^2$ to $H^1$.We also provide some partial results concerning the scattering of the first order moments, as well as a short proof via lens transform of a classical result due to Tsutsumi and Cazenave-Weissler on the scattering in $Σ$.

math.AP↗
arXiv2021

Growth of Sobolev Norms for 2d NLS with harmonic potential

We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the $2d$ cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s=2k, $k\in \mathbb{N}$, is $t^{2(s-1)/3+\varepsilon}$. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.

math.AP↗
arXiv2019

Transport of gaussian measures by the flow of the nonlinear Schrödinger equation

We prove a new smoothing type property for solutions of the 1d quintic Schrödinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.

math.AP↗
arXiv2016

Existence and Stability of standing waves for supercritical NLS with a Partial Confinement

We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.

math.AP↗
arXiv2012

Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation

Inspired by the work of Zhidkov on the KdV equation, we perform a construction of weighted gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. The resulting measures are supported by Sobolev spaces of increasing regularity. We also prove a property on the support of these measures leading to the conjecture that they are indeed invariant by the flow of the Benjamin-Ono equation.

math.AP↗
arXiv2009

On the orbital stability for a class of nonautonomous NLS

Following the original approach introduced by T. Cazenave and P.L. Lions in \cite{CaLi} we prove the existence and the orbital stability of standing waves for the following class of NLS: \label{intr1} i\partial_t u+ Δu - V(x) u + Q(x) u|u|^{p-2}=0, \hbox{} (t,x) \in \R\times \R^n, \hbox{} 2 λ_0\}\in (0,\infty)$ for a suitable $λ_0>0$. The main point is the analysis of the compactness of minimiziang sequences to suitable constrained minimization problems related to \eqref{intr1} and \eqref{intr2}.

math-ph↗