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Benoît Grébert

Publications and source records attributed to Benoît Grébert.

At least 19 recordsLinked to original sources
arXiv2026

Inverse Spectral Analysis of Singular Radial AKNS Operators

We study an inverse spectral problem for singular AKNS operators based on spectral data associated with two distinct values of the effective angular momentum parameter $κ\,$. Our main focus is the local inverse problem near the zero potential. For the pairs $(κ_1,κ_2)=(0,1)$, $(1,2)$ and $(0,3)\,$, we establish local uniqueness. For $(0,2)\,$, we prove that the Fréchet differential of the spectral map at the origin is injective, while the question whether its range is closed remains open.

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arXiv2026

On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta

We consider an inverse spectral problem for radial Schrödinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~$\ell$ satisfying a Müntz-type condition uniquely determines the potential. Next, in a neighborhood of the zero potential, we prove local uniqueness from two Dirichlet spectra associated with distinct angular momenta in the cases \((\ell_1,\ell_2) = (0,1)\,, \ (1,2)\) and \((0,3)\)\,. Our approach relies on an explicit analysis of the associated singular differential equation, combined with the classical Kneser--Sommerfeld formula. These results sharpen a theorem of Carlson-Shubin~(1994) and confirm, in the linearized setting and for these configurations, a conjecture originally formulated by Rundell and Sacks~(2001).

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arXiv2025

Almost global existence for Hamiltonian PDEs on compact manifolds

We prove an abstract result of almost global existence of small solutions to semi-linear Hamiltonian partial differential equations satisfying very weak non resonance conditions and basic multilinear estimates. Thanks to works by Delort--Szeftel, these assumptions turn out to typically hold for Hamiltonian PDEs on any smooth compact boundaryless Riemannian manifold. As a main application, we prove the almost global existence of small solutions to nonlinear Klein--Gordon equations on such manifolds: for almost all mass, any arbitrarily large $r$ and sufficiently large $s$, solutions with initial data of sufficiently small size $\varepsilon \ll 1$ in the Sobolev space $H^s \times H^{s-1}$ exist and remain in $H^s \times H^{s-1}$ for polynomial times $|t| \leq \varepsilon^{-r}$. This is the first result of almost global existence without specific assumptions on the compact manifold. We also apply this abstract result to nonlinear Schr{ö}dinger equations close to ground states and nonlinear Klein--Gordon equations on $\mathbb{R}^d$ with positive quadratic potentials.

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arXiv2025

On almost periodic solutions to NLS without external parameters

In this note, we present a result established in [BGR24] where we prove that nonlinear Schrodinger equations on the circle, without external parameters, admit plenty of infinite dimensional non resonant invariant tori, or equivalently, plenty of almost periodic solutions. Our aim is to propose an extended sketch of the proof, emphasizing the new points which have enabled us to achieve this result.

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arXiv2024

Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$

In this paper, we succeed in integrating Strichartz estimates (encoding the dispersive effects of the equations) in Birkhoff normal form techniques. As a consequence, we deduce a result on the long time behavior of quintic NLS solutions on the circle for small but very irregular initial data (in $H^s$ for $s > 2/5$). Note that since $2/5 < 1$, we cannot claim conservation of energy and, more importantly, since $2/5 < 1/2$, we must dispense with the algebra property of $H^s$. This is the first dynamical result where we use the dispersive properties of NLS in a context of Birkhoff normal form.

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arXiv2023

Exponential stability of solutions to the Schr{ö}dinger-Poisson equation

We prove an exponential stability result for the small solutions of the Schr{ö}dinger-Poisson equation on the circle without exterior parameters in Gevrey class. More precisely we prove that for most of the initial data of Gevrey-norm smaller than $\varepsilon$ small enough, the solution of the Schr{ö}dinger-Poisson equation remains smaller than $2\varepsilon$ for times of order $exp(α|\log \varepsilon|^2 / \log |\log \varepsilon|)$. We stress out that this is the optimal time expected for PDEs as conjectured by Jean Bourgain in [Bou04].

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arXiv2022

Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity

We are interested in the long time behavior of solutions of the nonlinear Schr{ö}dinger equation on the $d$-dimensional torus in low regularity, i.e. for small initial data in the Sobolev space $H^{s_0}(\mathbb T^d)$ with $s_0>d/2$. We prove that, even in this context of low regularity, the $H^s$-norms, $s\geq 0$, remain under control during times, $T_\varepsilon= \exp \big(-\frac{|\log\varepsilon|^2}{4\log|\log\varepsilon|} \big)$, exponential with respect to the initial size of the initial datum in $H^{s_0}$, $\|u(0)\|_{H^{s_0}}=\varepsilon$. For this, we add to the linear part of the equation a random Fourier multiplier in $\ell^\infty(\mathbb Z^d)$ and show our stability result for almost any realization of this multiplier. In particular, with such Fourier multipliers, we obtain the almost global well posedness of the nonlinear Schr{ö}dinger equation on $H^{s_0}(\mathbb T^d)$ for any $s_0>d/2$ and any $d\geq1$.

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arXiv2022

Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori

We consider quasi-linear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time $\varepsilon^{-2}$. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order $O(\varepsilon^{-4})$, in the Klein-Gordon case, we prove that the solutions exist at least for a time of order $O(\varepsilon^{-{8/3}^{-}})$ as soon as $d\geq3$. Regarding the Klein-Gordon equation, our result presents novelties also in the case of semi-linear perturbations: we show that the lifespan is at least of order $O(\varepsilon^{-{10/3}^-})$, improving, for cubic non-linearities and $d\geq4$, the general results in [17,24].

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arXiv2021

Discrete pseudo-differential operators and applications to numerical schemes

We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the fundamental property that the commutator of two discrete operators gains one order of regularity. We show that standard differential operators acting on periodic functions, finite difference operators and fully discrete pseudo-spectral methods fall into this class of discrete pseudo-differential operators. As examples of practical applications, we revisit standard error estimates for the convergence of splitting methods, obtaining in some Hamiltonian cases no loss of derivative in the error estimates, in particular for discretizations of general waves and/or water-waves equations. Moreover, we give an example of preconditioner constructions inspired by normal form analysis to deal with the similar question for more general cases.

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arXiv2021

Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2

We describe the long time behavior of small non-smooth solutions to the nonlinear Klein-Gordon equations on the sphere S^2. More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved for times of order $ε$^--r , where r >> 1 is an arbitrarily large number and $ε$ << 1 is the norm of the initial datum in the energy space H^1 x L^2. Roughly speaking, it means that, in order to exchange energy, modes have to oscillate at the same frequency. The proof relies on new multilinear estimates on Hamiltonian vector fields to put the system in Birkhoff normal form. They are derived from new probabilistic bounds on products of Laplace eigenfunctions that we obtain using Levy's concentration inequality.

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arXiv2021

Birkhoff normal forms for Hamiltonian PDEs in their energy space

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that, to exchange energy, modes have to oscillate at the same frequency. Contrary to the previous existing results, we do not require the solutions to be especially smooth. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d = 1 and nonlinear Schr{ö}dinger equations in dimension d $\le$ 2.

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arXiv2020

Long-time existence for semi-linear beam equations on irrational tori

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n -- 1 with n $\ge$ 3 and d $\ge$ 2. If $ε$ $\ll$ 1 is the size of the initial datum, we prove that the lifespan T$ε$ of solutions is O($ε$ --A(n--2) --) where A $\not\equiv$ A(d, n) = 1 + 3 d--1 when n is even and A = 1 + 3 d--1 + max(4--d d--1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain T$ε$ = O($ε$ --6 --), much better than O($ε$ --1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of $\sqrt$ $Δ$ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.

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arXiv2020

Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations

We provide an accurate description of the long time dynamics of the solutions of the generalized Korteweg-De Vries (gKdV) and Benjamin-Ono (gBO) equations on the one dimension torus, without external parameters, and that are issued from almost any (in probability and in density) small and smooth initial data. We stress out that these two equations have unbounded nonlinearities. In particular, we prove a long-time stability result in Sobolev norm: given a large constant r and a sufficiently small parameter $ε$, for generic initial datum u(0) of size $ε$, we control the Sobolev norm of the solution u(t) for times of order $ε$^{--r}. These results are obtained by putting the system in rational normal form : we conjugate, up to some high order remainder terms, the vector fields of these equations to integrable ones on large open sets surrounding the origin in high Sobolev regularity.

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arXiv2019

Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential

In this article we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than $1/2$. As far as we know, this is the first reducibility results for an unbounded perturbation of a linear system which is not integrable.

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arXiv2019

Reducibility of Schrödinger equation on the sphere

In this article we prove a reducibility result for the linear Schrödinger equation on the sphere $\mathbb{S}^{n}$ with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order strictly less than 1/2 and satisfying some parity condition. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. We notice that our result does not requires the use of the pseudo-differential calculus.

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arXiv2016

KAM for the nonlinear beam equation

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+Δ^2 u+m u + \partial_u G(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in \ { \mathbb{T}}^d, \qquad \qquad (*) $$ where $G(x,u)=u^4+ O(u^5)$. Namely, we show that, for generic $m$, many of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{G=0}$, written as the system $$ u_t=-v,\quad v_t=Δ^2 u+mu, $$ persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of $(*)$. If $d\ge2$, then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way.

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arXiv2016

On reducibility of Quantum Harmonic Oscillator on $\mathbb{R}^d$ with quasiperiodic in time potential

We prove that a linear d-dimensional Schr{ö}dinger equation on $\mathbb{R}^d$ with harmonic potential $|x|^2$ and small t-quasiperiodic potential $i\partial\_t u -- Δu + |x|^2 u + εV (tω, x)u = 0, x \in \mathbb{R}^d$ reduces to an autonomous system for most values of the frequency vector $ω\in \mathbb{R}^n$. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.

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