KAM theory for quasi-periodic equilibria in 1-D quasiperiodic media

TL;DR

This paper develops a KAM theory for quasi-periodic equilibria in 1D quasicrystals, providing a framework to validate solutions, analyze regularity, and compute analyticity breakdowns.

Contribution

It introduces an a-posteriori KAM theorem for quasi-periodic equilibria in 1D quasicrystals, with applications to validation and analysis of solutions.

Findings

Existence of true solutions near approximate ones under non-degeneracy conditions

Convergence of perturbative expansions for equilibria

Efficient algorithms for detecting analyticity breakdown

Abstract

We consider Frenkel-Kontorova models corresponding to 1 dimensional quasicrystals. We present a KAM theory for quasi-periodic equilibria. The theorem presented has an \emph{a-posteriori} format. We show that, given an approximate solution of the equilibrium equation, which satisfies some appropriate non-degeneracy conditions, then, there is a true solution nearby. This solution is locally unique. Such a-posteriori theorems can be used to validate numerical computations and also lead immediately to several consequences a) Existence to all orders of perturbative expansion and their convergence b) Bootstrap for regularity c) An efficient method to compute the breakdown of analyticity. Since the system does not admit an easy dynamical formulation, the method of proof is based on developing several identities. These identities also lead to very efficient algorithms.