- 11-56 Xifeng Su, Rafael de la Llave
- KAM theory for quasi-periodic equilibria in 1-D quasiperiodic media
Apr 11, 11
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Abstract. We consider Frenkel-Kontorova models corresponding to 1 dimensional
We present a KAM theory for quasi-periodic equilibria. The theorem
presented has an 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.