Response solutions for arbitrary quasi-periodic perturbations with Bryuno frequency vector

We study the problem of existence of response solutions for a real-analytic one-dimensional system, consisting of a rotator subjected to a small quasi-periodic forcing. We prove that at least one response solution always exists, without any assumption on the forcing besides smallness and analyticity. This strengthens the results available in the literature, where generic non-degeneracy conditions are assumed. The proof is based on a diagrammatic formalism and relies on renormalisation group techniques, which exploit the formal analogy with problems of quantum field theory; a crucial role is played by remarkable identities between classes of diagrams.Comment: 30 pages, 12 figure

KAM theory in configuration space and cancellations in the Lindstedt series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for quasi-periodic solutions with Diophantine frequency vector converges. If one studies the Lindstedt series, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where KAM theorem is usually formulated, by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser's modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. We revisit Moser's theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates. The interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.Comment: 38 pages, 18 fugure

Quasi-periodic solutions for the forced Kirchhoff equation on Td\mathbb{T}^d

In this paper we prove the existence of small-amplitude quasi-periodic solutions with Sobolev regularity, for the $d$-dimensional forced Kirchhoff equation with periodic boundary conditions. This is the first result of this type for a quasi-linear equations in high dimension. The proof is based on a Nash-Moser scheme in Sobolev class and a regularization procedure combined with a multiscale analysis in order to solve the linearized problem at any approximate solution.Comment: arXiv admin note: text overlap with arXiv:1311.694

Almost-periodic solutions to the NLS equation with smooth convolution potentials

We consider the one-dimensional NLS equation with a convolution potential and a quintic nonlinearity. We prove that, for most choices of potentials with polynomially decreasing Fourier coefficients, there exist almost-periodic solutions in the Gevrey class with frequency satisfying a Bryuno non-resonance condition. This allows convolution potentials of class $C^p$, for any integer $p$: as far as we know this is the first result where the regularity of the potential is arbitrarily large and not compensated by a corresponding smoothing of the nonlinearity.Comment: 140 pages, 32 figure

A KAM result on compact Lie groups

We describe some recent results on existence of quasi-periodic solutions of Hamiltonian PDEs on compact manifolds. We prove a linear stability result for the non-linear Schr\"odinger equation in the case of $SU(2)$ and $SO(3)$.Comment: 20 pages, 1 figur

Almost-periodic Response Solutions for a forced quasi-linear Airy equation

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig-Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on ${\mathbb T}^\infty$

Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems

We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeroes of a suitable function, called the Melnikov function - by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.Comment: 60 pages, 16 figures. arXiv admin note: substantial text overlap with arXiv:1011.093