Travelling Waves And Monotonicity In Coupled Oscillators Systems With Irrational Mean Spacing

This paper is composed of three parts. We firstly discuss the infinite dimensional Frenkel-Kontorova modelwhereΓ>0 is the damping coefficient, g is a periodic function, satisfying g(x+1)= g(x),∫01g(x)dx=0, where x∈R,b>0. K>0 measures the coupling strengths through position, F≥0 indicates the external driving force. What we are concerned with is the existence of travelling waves of the form Xj(t)=f(jw+vt), j∈Z, where f is the waveform function:R→R, satisfying f(t+1)=f(t)+1. w>0 and v>0 are mean spacing and average velocity of the particles, respectively. The main conclusion is as follows:There exists a Fd∈[0,1], such that the Frenkel-Kontorova model has a travelling wave solution provided F>Fd.Secondly, we study the monotonicity and average velocity of Frenkel-Kontorova system with irrational mean spacing w>0. The main conclusions are as follows: Assume Then the system is monotone and there exists a invariant subset Ew In fact, Ew is a Banach manifold which contains doubly infinite sequences with bounded width and mean spacing w>0. The average velocity v of the system exists independently of the initial conditions. Moreover, v is a nondecreasing and continuous function of F.Finally, we study a chain of coupled oscillators with strong dampingSimilar conclusions hold true under the overdamped condition-2β<Γ<2α/β.