Minimal Configurations in the Aubry-Mather Theory

  • Simeon Reich
  • Alexander J. Zaslavski
Part of the Developments in Mathematics book series (DEVM, volume 34)

Abstract

Chapter 10 is devoted to the Aubry-Mather theory applied to the famous Frenkel-Kontorova model, an infinite discrete model of solid-state physics related to dislocations in one-dimensional crystals. In this model a configuration of a system is a sequence of real numbers with indices from −∞ to +∞. We are interested in (h)-minimal configurations with respect to an energy function h. A configuration is called (h)-minimal if its total energy cannot be made less by changing its final states. Classical Aubry-Mather theory is concerned with finding and investigating h-minimal configurations with a given rotation number, where the function h is fixed. It implies that the set of all periodic h-minimal configurations of a rational rotation number p/q is totally ordered. Moreover, between any two neighboring periodic h-minimal configurations with rotation number p/q, there are (non-periodic) h-minimal heteroclinic connections having the same rotation number p/q. We consider a complete metric space of energy functions h equipped with a certain C 2 topology and show that for most energy functions in this space, there exist three different h-minimal configurations with rotation number p/q such that any other h-minimal configuration with the same rotation number p/q is a translation of one of these three.

Keywords

Energy Function Rational Number Dense Subset Nonnegative Function Rotation Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Simeon Reich
    • 1
  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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