August  2010, 27(3): 1147-1158. doi: 10.3934/dcds.2010.27.1147

Rotating modes in the Frenkel-Kontorova model with periodic interaction potential

1. 

Department of Mathematics, Suzhou University, Suzhou, 215006, China

Received  July 2009 Revised  January 2010 Published  March 2010

Employing a homotopy argument and the Leray-Schauder degree theory, we show the existence of rotating modes for the Frenkel-Kontorova model with periodic interaction potential. The solutions describing rotating modes are periodic and called rotating oscillating solutions, in which the phase of a fixed rotator increases by $2\pi$ per period, while its neighbors oscillate with small amplitudes around their equilibrium positions. We also discuss a fundamental difference between the Frenkel-Kontorova model with periodic interaction potential and that with convex interaction potential by demonstrating the nonexistence of the rotating modes for the latter case.
Citation: Wen-Xin Qin. Rotating modes in the Frenkel-Kontorova model with periodic interaction potential. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1147-1158. doi: 10.3934/dcds.2010.27.1147
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