Related papers: KAM \`{a} la R
We extend weak KAM theory to Lagrangians that are defined only on the horizontal distribution of a sub-Riemannian manifold. The main tool is Tonelli's theorem which allows dispending on a Lagrangian dynamics.
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 prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with periodic boundary…
This is a short survey on Nekhoroshev theory, KAM theory, and Arnold's diffusion.
We consider Gevrey perturbations $H$ of a completely integrable Gevrey Hamiltonian $H_0$. Given a Cantor set $\Omega_\kappa$ defined by a Diophantine condition, we find a family of KAM invariant tori of $H$ with frequencies $\omega\in…
Based on quantitative ``{\sc kam} theory'', we state and prove two theorems about the continuation of maximal and whiskered quasi--periodic motions to slightly perturbed systems exhibiting proper degeneracy. Next, we apply such results to…
We consider one dimensional chains of interacting particles subjected to one dimensional almost-periodic media. We formulate and prove two KAM type theorems corresponding to both short-range and long-range interactions respectively. Both…
This paper presents new variants of the averaged alternating modified reflections (AAMR) method for the best approximation problem. Under a mild constraint qualification, we first show its weak convergence and then establish a convergence…
We give a simple proof of Kolmogorov's theorem on the persistence of a quasiperiodic invariant torus in Hamiltonian systems. The theorem is first reduced to a well-posed inversion problem (Herman's normal form) by switching the frequency…
This paper continues the discussion started in [CK19] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit `global' Arnold's KAM Theorem, which yields, in particular, the…
From the beginning of KAM theory, it was realized that its applicability to realistic problems depended on developing quantitative estimates on the sizes of the perturbations allowed. In this paper we present results on the existence of…
We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and "drift". Our goal is to construct quasi-periodic…
We prove that a system of coupled nonlinear Schr{\"o}dinger equations on the torus exhibits both stable and unstable small KAM tori. In particular the unstable tori are related to a beating phenomena which has been proved recently in [6].…
We prove the existence and stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian perturbations of KdV.
In discrete schemes, weak KAM solutions may be interpreted as approximations of correctors for some Hamilton-Jacobi equations in the periodic setting. It is known that correctors may not exist in the almost periodic setting. We show the…
A selfcontained proof of the KAM theorem in the Thirring model is discussed.
In this paper, we strengthen the splitting theorem proved in [14, 15] and provide a different approach using ideas from the weak KAM theory.
Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable…
Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data.
A method is suggested for treating those complicated physical problems for which exact solutions are not known but a few approximation terms of a calculational algorithm can be derived. The method permits one to answer the following rather…