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, (Formula Presented.) is driven from one end by a particle x₀. This particle undergoes a given, periodically perturbed, uniform motion x0(t) = 2at + 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 system (Formula Presented.) exist in the cases γ > γ₁ and γ₁ > γ > γ₂ 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 type (Formula Presented.) with k = 0 or 1 for general F and k arbitrary for F(x) = e^x (when k = 0, take γ₀ = ∞ and X₀ ≡ 0).