Related papers: KAM theorems and open problems for infinite dimens…

In this paper we apply symplectic algorithms to nearly integrable Hamiltonian system, and prove it can maintain lots of elliptic lower dimensional invariant tori. We are committed to consider the elliptic lower dimensional invariant tori…

In this paper we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems \begin{equation*} H=h(y)+f(x,y,t), \end{equation*} where $y\in D\subseteq\mathbb{R}^n$ with $D$ being a closed bounded domain,…

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…

Chow, Li and Yi in [2] proved that the majority of the unperturbed tori {\it on sub-manifolds} will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies…

As applications of Kadison's Pythageorean and carpenter's theorems, the Schur-Horn theorem, and Thompson's theorem, we obtain an extension of Thompsons theorem to compact operators and use these ideas to give a characterization of diagonals…

Short-ranged and line-gapped non-hermitian Hamiltonians have strong topological invariants given by an index of an associated Fredholm operator. It is shown how these invariants can be accessed via the signature of a suitable spectral…

We establish a correspondence between the semi-infinite and infinite Volterra lattices having a finite logarithmic Hamiltonian and certain classes of even probability measures. In doing so, we apply the inverse spectral theory of Jacobi…

In the Nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophatine critical point of a hamiltonian function. I construct a formalism for the UV-cutoff and prove a generalised KAM…

We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $\omega$ and a family of presymplectic mappings, if we find an embedded torus which is…

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderely, completely integrable systems, are charactrized by phase trajectories…

In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful…

We prove an analytic KAM-Theorem, which is used in [1], where the differential part of KAM-theory is discussed. Related theorems on analytic KAM-theory exist in the literature (e. g., among many others, [7], [8], [13]). The aim of the…

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 prove an abstract KAM theorem adapted to space-multidimensional hamiltonian PDEs with regularizing nonlinearities. It applies in particular to the singular perturbation problem studied in the first part of this work.

In 2004, F\'ejoz [D\'emonstration du 'th\'eor\'eme d'Arnold' sur la stabilit\'e du syst\`eme plan\'etaire (d'apr\`es M. Herman). Ergod. Th. & Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's [D\'emonstration d'un…

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,…

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of…

In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another…

Our understanding of the mechanisms governing the structure and secular evolution galaxies assume nearly integrable Hamiltonians with regular orbits; our perturbation theories are founded on the averaging theorem for isolated resonances. On…

We revisit a K-theoretical invariant that was invented by the first author some years ago for studying multiparameter bifurcation of branches of critical points of functionals. Our main aim is to apply this invariant to investigate…