Related papers: A Kam Theorem for Space-Multidimensional Hamiltoni…

Consider a sufficiently smooth nearly integrable Hamiltonian system of two and a half degrees of freedom in action-angle coordinates \[ H_\epsilon (\varphi,I,t)=H_0(I)+\epsilon H_1(\varphi,I,t), \varphi\in T^2,\ I\in U\subset R^2,\ t\in…

In this paper, we present two infinite-dimensional KAM theorems with frequency-preserving for a nonresonant frequency of Diophantine type or even weaker. To be more precise, under a nondegenerate condition for an infinite-dimensional…

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact…

This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM tori in a singly thermostated, 1 degree-of-freedom hamiltonian vector field. This result is applied to 4 important single thermostats…

Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstates of quantum mechanical Hamiltonians that heretofore were not believed to be obtainable by such methods. The novel feature of adaptive…

In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian having an invariant torus supporting…

We consider the $d$-dimensional nonlinear Schr\"odinger equation under periodic boundary conditions: $-i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is…

We discuss a Nash-Moser/ KAM algorithm for the construction of invariant tori for {\em tame} vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the…

We prove the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic…

We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is…

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…

Commutative $d$-torsion $K$-theory is a variant of topological $K$-theory constructed from commuting unitary matrices of order dividing $d$. Such matrices appear as solutions of linear constraint systems that play a role in the study of…

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…

We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear difference equations assuming a very general form of dichotomic behavior for the linear equation. The results obtained…

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a…

All the almost periodic solutions for non integrable PDEs found in the literature are very regular (at least $C^\infty$) and, hence, very close to quasi periodic ones. This fact is deeply exploited in the existing proofs. Proving the…

In this paper we prove the existence and the stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced Kirchoff equation with periodic boundary conditions. This is the first KAM result for a…

We prove that there is an invariant torus with given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for…

At the light of recent results in literature we review a conjecture formulated in Math. Phys. Electron. J. 1 (1995), paper 5, 1--13, about the mechanism of breakdown of invariant sets in KAM problems and the identification of the dominant…

We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero square-integrable harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on (1-forms on M)/Ker(d)?