Lei Zhang, Xifeng Su, Rafael de la Llave
Equilibrium quasi-periodic
configurations with resonant frequencies in quasi-periodic media
II: KAM theor
configurations with resonant frequencies in quasi-periodic media
II: KAM theory}
(276K, pdf)
ABSTRACT. We develop an a-posteriori KAM theory for the equilibrium equations for
quasi-periodic solutions in a quasi-periodic Frenkel-Kontorova
model when the frequency of the solutions resonates with the
frequencies of the substratum.
The KAM theory we develop is very different both in the methods
and in the conclusions from the more customary KAM theory for
Hamiltonian systems or from the KAM theory in quasi-periodic media
for solutions with frequencies Diophantine with respect to the frequencies of th
e media. The main difficulty is that we cannot use transformations (as in the Ha
miltonian case) nor Ward identities
(as in the case of frequencies Diophantine with those of the media).
The technique we use is to add an extra equation to make the linearization of th
e equilibrium equation factorize. This requires
an extra counterterm. We compare this phenomenon with other phenomena in KAM theory. It seems that this technique could be used in several other problems.
As a conclusion, we obtain that the perturbation expansions developed in the previous
paper [SuZL14] converge when the potentials are in a codimension one manifo
ld in a space of potentials.
The method of proof also leads to efficient (low storage requirements and
low operation count) algorithms to compute the quasi-periodic solutions.