Related papers: Quantitative KAM normal forms and sharp measure es…

In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional…

We study a quantum analogue of the iterative perturbation theory by Kolmogorov used in the proof of the Kolmogorov-Arnold-Moser (KAM) theorem. The method is based on sequent canonical transformations with a "running" coupling constant $…

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…

Perturbations due to round-off errors in computer modeling are discontinuous and therefore one cannot use results like KAM theory about smooth perturbations of twist maps. We elaborate a special approximation scheme to construct two smooth…

In this paper, we investigate analytic divergence-free vector fields and vector fields admitting a Jacobi multiplier on $n$-dimensional Riemannian manifolds. We first introduce a functional acting on the space of divergence-free vector…

In this paper we prove an infinite dimensional KAM theorem, in which the assumptions on the derivatives of perturbation in \cite{GT} are weakened from polynomial decay to logarithmic decay. As a consequence, we apply it to 1d quantum…

A selfcontained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the ``strong diophantine property'' hypothesis used in previous papers. Keywords: \it KAM, invariant tori, classical mechanics, perturbation…

In this work we consider the KAM renormalizability problem for small pseudodifferential perturbations of the semiclassical isochronous transport operator with Diophantine frequencies on the torus. Assuming that the symbol of the…

We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The…

In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In…

We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser…

We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) $O(\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with…

Computationally efficient, structure-preserving reduced-order methods are developed for the Korteweg de Vries (KdV) equations in Hamiltonian form. The KdV equation is discretized in space by finite differences. The resulting skew-gradient…

We introduce an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible Schr\"odinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we find the existence of quasi-periodic…

For a fixed frequency vector $\omega \in \mathbb{R}^2 \, \setminus \, \lbrace 0 \rbrace$ obeying $\omega_1 \omega_2 < 0$ we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate…

In this paper we present efficient algorithms for the computation of several invariant objects for Hamiltonian dynamics. More precisely, we consider KAM tori (i.e diffeomorphic copies of the torus such that the motion on them is conjugated…

In this paper we reconsider the original Kolmogorov normal form algorithm with a variation on the handling of the frequencies. At difference with respect to the Kolmogorov approach, we do not keep the frequencies fixed along the…

We consider the $ C^1 $ linearization of a perturbed vector field $ \omega+P $ over the infinite-dimensional torus $ \mathbb{T}^\infty $, and determine the sharp regularity requirement of the perturbation $ P $ conjugating the unperturbed…

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori…

We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer…