Reducibility for a fast driven linear Klein-Gordon equation

TL;DR

This paper proves that a linear Klein-Gordon equation with fast oscillating quasi-periodic forcing can be transformed into a simpler, time-independent form using a KAM scheme, under certain frequency conditions.

Contribution

It introduces a novel reducibility result for the Klein-Gordon equation with fast oscillations, employing a new type of Melnikov conditions for convergence.

Findings

Equation is conjugated to a time-independent diagonal form

Requires external frequency to be large and from a large measure Cantor set

Uses a two-step process with a preliminary transformation and KAM scheme

Abstract

We prove a reducibility result for a linear Klein-Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving, however we require it to be fast oscillating. In particular, provided that the external frequency is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time independent, diagonal one. We achieve this result in two steps. First, we perform a preliminary transformation, adapted to fast oscillating systems, which puts the original equation in a perturbative setting. Then we show that this new equation can be put to constant coefficients by applying a KAM reducibility scheme, whose convergence requires a new type of Melnikov conditions.