Related papers: A Birkhoff Normal Form Theorem for Partial Differe…

We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in the sense that nearby solutions remain close to the torus…

We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results…

In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to…

We consider Kirchhoff equations for vibrating bodies in any dimension in presence of a time-periodic external forcing with period 2pi/omega and amplitude epsilon, both for Dirichlet and for space-periodic boundary conditions. We prove…

We consider the quintic nonlinear Schr{\"o}dinger on the circle. By applying a Birkhoff procedure and a KAM theorem, we exihibit a three dimension invariant torus that is linearly unstable. In comparison, we also prove that two dimensional…

We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillates in a chaotic way on certain long time scales. The chaoticity is encoded in the time between oscillations of the norm,…

We give a necessary and sufficient condition for strong stability of low dimensional Hamiltonian systems, in terms of the iterates of a closed orbit and the Conley-Zehnder index. Applications to Mathieu equation and stable harmonic…

We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schr\"odinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove…

We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\leq n$. The result we get ensures the persistence of an…

For dispersive Hamiltonian partial differential equations of order 2N+1, N integer, there are two criteria to analyse to examine the stability of small-amplitude, periodic travelling wave solutions to high-frequency perturbations. The first…

We derive the Whitham equations from the water waves equations in the shallow water regime using two different methods, thus obtaining a direct and rigorous link between these two models. The first one is based on the construction of…

In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed…

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term…

Using a Birkhoff normal form transform to impede mode transfer in a finite "barrier", we prove localization of arbitrary $\ell^2$ data for polynomially long time for the nonlinear quasi-periodic Schr\"odinger equation on $\mathbb Z^d$.

We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of…

We present a new, short proof of the increased regularity obtained by solutions to uniformly parabolic partial differential equations. Though this setting is fairly introductory, our new method of proof, which uses a priori estimates, can…

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability…

In this work, we introduce a new space-time variational formulation of the second-order wave equation, where integration by parts is also applied with respect to the time variable, and a modified Hilbert transformation is used. For this…

In this paper we prove the existence of multiple periodic solutions (harmonic and subharmonic) for a class of planar Hamiltonian systems which include the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a…

We consider a space-time variational formulation of the second-order wave equation, where integration by parts is also applied with respect to the time variable. Conforming tensor-product finite element discretisations with piecewise…