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Reducibility for a fast-driven linear Klein–Gordon equation

  • L. Franzoi
  • A. MasperoEmail author
Article
  • 22 Downloads

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 moves 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.

Keywords

Reducibility KAM theory Fast driving potential Klein–Gordon equation 

Mathematics Subject Classification

35L10 37K55 

Notes

Acknowledgements

We thank Dario Bambusi, Massimiliano Berti, Roberto Feola, Matteo Gallone and Vieri Mastropietro for many stimulating discussions. We were partially supported by Prin-2015KB9WPT and Progetto GNAMPA - INdAM 2018 “Moti stabili ed instabili in equazioni di tipo Schrödinger”.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.International School for Advanced Studies (SISSA)TriesteItaly

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