Related papers: An Infinite-dimensional KAM Theorem with Normal De…
The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear,…
Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K\"ahler setting, we can then deduce a…
In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action…
This paper is an announcement of a result followed with explanations of some ideas behind. The proofs will appear elsewhere. Our goal is to construct a Hamiltonian perturbation of any completely integrable Hamiltonian system with $2n$…
In view of growing interest in tensor modes and their possible detection, we clarify the definition of tensor modes up to 2nd order in perturbation theory within the Hamiltonian formalism. Like in gauge theory, in cosmology the Hamiltonian…
We prove the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of $\mathbf{R}^3$ where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma…
The paper consists of two sections. In Section 1, we give a short review of KAM theory with an emphasis on Whitney smooth families of invariant tori in typical Hamiltonian and reversible systems. In Section 2, we prove a KAM-type result for…
We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which…
We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely…
The question of the total measure of invariant tori in analytic, nearly--integrable Hamiltonian systems is considered. In 1985, Arnol'd, Kozlov and Neishtadt, in the Encyclopaedia of Mathematical Sciences \cite{AKN1}, and in subsequent…
Noncommutatively deformed geometries, such as the noncommutative torus, do not exist generically. I showed in a previous paper that the existence of such a deformation implies compatibility conditions between the classical metric and the…
In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an {\em a posteriori} approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for…
Degeneracy is a method to accommodate exact, low energy vacuum states in scalar-tensor gravitational models despite the presence of an arbitrarily large vacuum energy. However, this approach requires very particular combinations of scalar…
In this paper, we prove the existence of full dimensional tori for $d$-dimensional nonlinear Schr$\ddot{\mbox{o}}$dinger equation with periodic boundary conditions \begin{equation*}\label{L1} \sqrt{-1}u_{t}+\Delta u+V*u\pm\epsilon…
In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover,…
In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + g(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in {\mathbb T}^d, \qquad \qquad (*) $$ where $g(x,u)=4u^3+ O(u^4)$. Namely,…
In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem…
We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics…
We review a recent generalization of Normal Form Theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main difference with the standard case relies on the non…
In this paper, we prove the existence of full dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via KAM theory.