Forced lattice vibrations: Part II

This is the second part of a two‐part series on forced lattice vibrations in which a semi‐infinite lattice of one‐dimensional particles { x n } n ≧1 ,is driven from one end by a particle x 0 . This particle undergoes a given, periodically perturbed, uniform motion x0 ( t ) = 2a t + h ( yt ) where a and γ are constants and h (·) has period 2π. Results and notation from Part I are used freely and without further comment. Here the authors prove that sufficiently ample families of traveling‐wave solutions of the doubly infinite systemexist in the cases γ > γ 1 and γ 1 > γ > γ 2 for general restoring forces F. In the case with Toda forces, F(x) = e x , the authors prove that sufficiently ample families of traveling‐wave solutions exist for all k , γ k > γ > γ k +1 . By a general result proved in Part I, this implies that there exist time‐periodic solutions of the driven system (i) with k ‐phase wave asymptotics in n of the typewith k = 0 or 1 for general F and k arbitrary for F ( x ) = e x (when k = 0, take γ 0 = ∞ and X 0 ≡ 0).