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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 this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics \Omega_{i}(\xi)=i^d+o(i^{d})+o(i^{\delta}), where $d>0, \delta<0$, which can be applied to a large class…

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary…

{\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…

In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schr\"odinger equation on the two dimensional torus. We prove that these…

We consider models of one-dimensional chains of non-nearest neighbor and many-body interacting particles subjected to quasi-periodic media. We extend the results in \cite{12Su&delaLlavelongrange} from analytic to Gevrey regularity…

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure…

The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract…

We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\leq n$. The result we get ensures the persistence of an…

By constructing an infinite dimensional KAM theorem of the normal frequencies being dense at finite-point, we show that some shallow water equations such as Benjamin-Bona-Mahony equation and the generalized $d$-Dim. Pochhammer-Chree…

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…

We consider one dimensional chains of interacting particles subjected to one dimensional almost-periodic media. We formulate and prove two KAM type theorems corresponding to both short-range and long-range interactions respectively. Both…

In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed…

A generalized Benjamin-Bona-Mahony (gBBM) equation subject to the periodic boundary condition is studied in this paper. Based on a new infinite dimensional Kolomogorov-Arnold-Moser (KAM) theorem with normal frequencies of finite…

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…

The paper consists of two sections. In Section 1, we give a short review of KAM theory with an emphasis on Whitney smooth families of invariant tori in typical Hamiltonian and reversible systems. In Section 2, we prove a KAM-type result for…

We prove the existence of quasi-periodic, small amplitude, solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities we also obtain the linear stability of the…

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…

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 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,…