The Analyticity Breakdown for Frenkel-Kontorova Models in Quasi-periodic Media: Numerical Explorations
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We study numerically the “analyticity breakdown” transition in 1-dimensional quasi-periodic media. This transition corresponds physically to the transition between pinned down and sliding ground states. Mathematically, it corresponds to the solutions of a functional equation losing their analyticity properties.
We implemented some recent numerical algorithms that are efficient and backed up by rigorous results so that we can compute with confidence even close to the breakdown.
We have uncovered several phenomena that we believe deserve a theoretical explanation: (A) The transition happens in a smooth surface. (B) There are scaling relations near breakdown. (C) The scaling near breakdown is very anisotropic. Derivatives in different directions blow up at different rates.
Similar phenomena seem to happen in other KAM problems.
KeywordsQuasi-periodic solutions Quasi-crystals Hull functions KAM theory
Part of this work was done while the authors were affiliated with Univ. of Texas. We thank S. Hernandez and X. Su for many discussions about this problem and about numerical issues. We also thank the Center for Nonlinear Analysis (NSF Grants No. DMS-0405343 and DMS-0635983), where part of this research was carried out.
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