Forced lattice vibrations: Part I

This is the First 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) = _at + _h(_yt), where a and γ are constants and h(·) has period 2π. For a wide variety of restoring forces F (i.e., F′ > 0), numerical calculations indicate the existence of a sequence of thresholds γ₁ = γ₁(a, h, F) > γ₂ = γ₂(a,h,F) > … > γ_k = γ_k(a,h,F) > …, γ_k → 0, as k → ∞. If γ_k > γ > γ_k+1, a k‐phase wave that is well described by the wave form, (Formula Presented.) emerges and travels through the lattice. The goal of this series is to describe the emergence and calculate some properties of these wave forms. In Part I the authors first consider the case where F(x) = e^x (i.e., Toda forces) but h is arbitrary, and show how to compute a basic diagnostic (see J(λ), formula (1.26)) for the system in terms of the solution of an associated scalar Riemann‐Hilbert problem, once a certain finite set of numbers is known. In another direction, the authors consider the case where F(x) is restoring but arbitrary, and h is small. Here the authors prove a general result, asserting that if there exists a sufficiently ample family of traveling‐wave solutions of the doubly infinite lattice, (Formula Presented.) then it is possible to construct time‐periodic k‐phase wave solutions with asymptotics in n of type (iii) for the driven system (i). In Part II, the authors prove that sufficiently ample families of traveling‐wave solutions of the system (iv) 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.