arXiv subjects

Benoît Grébert

Publications and source records attributed to Benoît Grébert.

25 records · Page 2Linked to original sources

arXiv2016

KAM for the Klein Gordon equation on $\mathbb S^d$

Recently the KAM theory has been extended to multidimensional PDEs. Nevertheless all these recent results concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the structure of the resonant sets is quite simple. In the present paper, we consider an important physical example that do not fit in this context: the Klein Gordon equation on $\mathbb S^d$. Our abstract KAM theorem also allow to prove the reducibility of the corresponding linear operator with time quasiperiodic potentials.

math.AP↗

arXiv2015

Modified scattering for the cubic Schr{ö}dinger equation on product spaces: the nonresonant case

We consider the cubic nonlinear Schr{ö}dinger equation on the spatial domain $\mathbb{R}\times \mathbb{T}^d$, and we perturb it with a convolution potential. Using recent techniques of Hani-Pausader-Tzvetkov-Visciglia, we prove a modified scattering result and construct modified wave operators, under generic assumptions on the potential. In particular, this enables us to prove that the Sobolev norms of small solutions of this nonresonant cubic NLS are asymptotically constant.

math.AP↗

arXiv2013

Dynamics of Klein-Gordon on a compact surface near an homoclinic orbit

We consider the Klein-Gordon equation on a Riemannian surface which is globally well-posed in the energy space. This equation has an homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we show that there are many solutions which stay close to this homocline during all times. We point out that the solutions we construct are not small.

math.AP↗

arXiv2013

Stability of large periodic solutions of Klein-Gordon near a homoclinic orbit

We consider a Klein-Gordon equation (KG) on a Riemannian compact surface, for which the flow lets invariant the two dimensional space the solutions independent of the space variable. It turns out that in this invariant space, there is a homoclinic orbit to the origin, and a family of periodic solutions inside the loops of the homoclinic orbit. In this paper we study the stability of these periodic orbits under the (KG) flow, i.e. when turning on the nonlinear interaction with the non stationary modes. By a shadowing method, we prove that around the periodic orbits, solutions stay close to them during a time of order $(\log a)^2$, where $a$ is the distance between the periodic orbit considered and the homoclinic orbit.

math.AP↗

arXiv2012

Beating effects in cubic Schrödinger systems and growth of Sobolev norms

We consider a system of coupled cubic Schrödinger equations. We prove that there exists a beating effect, i.e. an energy exchange between different modes. This construction may be transported to the linear time-dependent Schrödinger equation: we build solutions such that their Sobolev norms grow logarithmically. All these results are stated for large but finite times.

math.AP↗

arXiv2011

KAM for the quantum harmonic oscillator

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.

math.AP↗

arXiv2009

Normal form theory for the NLS equation

We consider the nonlinear Schrodinger equation with a cubic nonlinearity on the circle, which is known to represent an integrable Hamiltonian system. We construct a global coordinate systems, which puts this Hamiltonian into standard normal form, so that it depends only on the actions of the system.

math.AP↗