Related papers: KAM for reversible derivative wave equations

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…

Recently R\"ussmann proposed a new new variant of KAM theory based on a slowly converging iteration scheme. It is the purpose of this note to make this scheme accessible in an even simpler setting, namely for analytic perturbations of…

In this paper, we prove that the Sobolev norm of solutions of the linear wave equation with unbounded perturbations of order one stay bounded for the all time. The main proof is based on the KAM reducibility of the linear wave equation. To…

We consider extended CMV matrices with analytic quasi-periodic Verblunsky coefficients with Diophantine frequency vector in the perturbatively small coupling constant regime and prove the analyticity of the tongue boundaries. As a…

We consider the nonlinear Schr\"{o}dinger equation of degree five on the circle $\mathbb{S}^1 = \mathbb{R}/2\pi$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system…

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\phi(u)$, with $\phi(u)=\Phi u^{3} + O(u^{5})$ odd and analytic, $\Phi\neq0$, and we construct small amplitude periodic solutions with frequency…

In this paper, we prove that the existence of globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking. We first introduce a new set of independent and dependent variables in…

Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we…

We show that every (invertible, or noninvertible) minimal Cantor system embeds in $\mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

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…

We prove existence of weak solutions (in the probabilistic sense) for a general class of stochastic semilinear wave equations on bounded domains of $R^d$ driven by a possibly discontinuous square integrable martingale.

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We…

We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a…

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in…

We consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. Under the null condition for such systems, the global existence of small amplitude solutions is known. In…

This note is a supplement with a new result to the review paper by Takamura [13] on nonlinear wave equations in one space dimension. We are focusing here to the long-time existence of classical solutions of semilinear wave equations in one…

We present a novel approach for deriving KAM-type linearization theorems directly -- and almost immediately -- from the existence of the stable foliation for a renormalization operator. We give a few illustrations in dynamics in one and…

We propose compact finite difference schemes to solve the KP equations $u\_t + u\_{xxx} + u^p u\_x + $\lambda$ \partial^{--1}\_x u\_{yy} = 0$. When $p = 1$, this equation describes the propagation of small amplitude long waves in shallow…

This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyaponuv function, we prove that when the damping is linear, we can find initial…

In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type [8]. We reported conditions which guarantee convergence…