Related papers: Towards sharp regularity: Full-dimensional tori in…
In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are…
In this paper, we present two infinite-dimensional Kolmogorov theorems based on non-resonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a Legendre-type nondegeneracy condition for an…
{\bf Abstract}: The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ subject to certain boundary conditions by means of…
A method via the KAM technique is introduced to study the existence of invariant tori and quasiperiodic solutions for impulsive Duffing-type equations with time period 1. Basing on several planar symplectic homeomorphisms and some estimates…
In this paper, we investigate the sharp regularity properties of a special weighted Sobolev space defined on the $ n $-dimensional torus, which is of independent interest. As a key application, we show that for almost all $ n $-dimensional…
Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the…
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…
We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector $\o_0$ is always accumulated by invariant complex analytic…
In this paper we present an a-posteriori KAM theorem for the existence of an $(n-d)$-parameters family of $d$-dimensional isotropic invariant tori with Diophantine frequency vector $\omega\in \mathbb R^d$, of type $(\gamma,\tau)$, for $n$…
In the present paper, we will discuss the following non-degenerate Hamiltonian system \begin{equation*} H(\theta,t,I)=\frac{H_0(I)}{\varepsilon^{a}}+\frac{P(\theta,t,I)}{\varepsilon^{b}}, \end{equation*} where…
We show that an analytic invariant torus $\cT_0$ with Diophantine frequency $\o_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\cT_0$ satisfies a R\"ussmann transversality…
In this paper, we study the Hamiltonian systems $ H\left( {y,x,\xi ,\varepsilon } \right) = \left\langle {\omega \left( \xi \right),y} \right\rangle + \varepsilon P\left( {y,x,\xi ,\varepsilon } \right) $, where $ \omega $ and $ P $ are…
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…
In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schr\"odinger equation \begin{eqnarray}\label{maineq0} \mathbf{i}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^{4}u=0,\…
In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + g(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in {\mathbb T}^d, \qquad \qquad (*) $$ where $g(x,u)=4u^3+ O(u^4)$. Namely,…
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…
In this paper, we study the following nonlinear Schr\"odinger equation \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u=0,\ x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}, \end{eqnarray} where $V*$ is the Fourier…
The parametric equations of KAM tori for a quasi integrable system, are shown to be one point Schwinger functions of a suitable euclidean quantum field theory on the torus. KAM theorem is equivalent to a ultraviolet stability theorem. A…
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…
In this paper, we prove a KAM theorem in a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with $n$ degrees of freedom that depend periodically or quasi-periodically (QP) on…