Related papers: Quasi-T\"oplitz functions in KAM theorem

We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser…

In this paper, we investigate the almost-periodic solutions for the one-dimensional nonlinear Klein-Gordon equation within the non-relativistic limit under periodic boundary conditions. Specifically, by employing the method introduced in…

In this paper, one-dimensional (1D) nonlinear wave equations $u_{tt} -u_{xx}+V(x)u =f(u)$, with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function…

We provide a symplectic reduction of a partially integrable Hamiltonian system to a completely integrable one. The KAM theorem is applied to this reduced completely integrable Hamiltonian system. Its KAM perturbation generates a…

We construct time quasi-periodic solutions to the energy supercritical nonlinear Schr\"odinger equations on the torus in arbitrary dimensions. This introduces a new approach, which could have general applicability.

In this paper, we investigate the existence of KAM tori for an infinite dimensional Hamiltonian system with finite number of zero normal frequencies. By constructing a constant quantity we show that, for "most" frequencies in the sense of…

{\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…

We consider a damped/driven cubic NLS equation on a torus under the limit when first the amplitude of solutions goes to zero and then the period of the torus goes to infinity. We suggest another proof of the kinetic approximation for the…

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 study the persistence and remaining regularity of KAM invariant torus under sufficiently small perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus of…

In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schr\"odinger equation with periodic boundary conditions \begin{equation*}\label{L1}…

We investigate the dynamics of the quasi-periodic swing equations from the perspective of weak KAM theory. To this end, we firstly study a class of Hamiltonian systems. We obtain that the limit $u$, which derived from convergence of a…

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain (2005) on the quintic NLS, we propose a novel…

In the present work we explore the potential of models of the discrete nonlinear Schr\"odinger (DNLS) type to support spatially localized and temporally quasiperiodic solutions on top of a finite background. Such solutions are rigorously…

It is widespread since the beginning of KAM Theory that, under "sufficiently small" perturbation, of size $\epsilon$, apart a set of measure $O(\sqrt{\epsilon})$, all the KAM Tori of a non-degenerate integrable Hamiltonian system persist up…

We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, implementing a Backlund transformation of SL equation. In this…

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 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,…

This paper focuses on the problem of quasi-periodic solutions for multi-dimensional quasi-linear Schr\"odinger equation. To address the challenge of unbounded perturbations caused by quasi-linear terms in the equation, we define the…

We study non-linear functionals, including quasi-linear functionals, p-conic quasi-linear functionals, d-functionals, r-functionals, and their relationships to deficient topological measures and topological measures on locally compact…