Sliding States In Damped Frenkel-Kontorova Model

The Frenkel-Kontorova model (FK model) is one of the fundamental models incoupling oscillator systems. It has a wide range of applications in nonlinear physics.The sliding states play an important role in the FK model.In this paper, we establish the existence of uniform sliding states for the FK modelusing topological degree method. For the damped nonlinear coupling monatomic chain,in which all particles have the same mass, we construct a compact homotopic mapping,then convert the existence of sliding states to that of fixed points of a compact mappingbased on the invariance property about the Leray-Schauder degree for compact homo-topic mappings. Then we use the index theorem of isolated zero to gain the existenceof the fixed point. For the damped linear coupling diatomic chain, in which the massfor odd sites is diferent from that for even sites, we obtain the existence of slidingstates applying the Schauder fixed point theorem, after converting to the question ofthe existence of fixed points for a compact mapping. Finally we discuss the range ofparameters for sliding states to exist.