Forced lattice vibrations: Part II

Percy Deift, Thomas Kriecherbauer, Stephanos Venakides

Research output: Contribution to journalArticle

Abstract

This is the second part of a two‐part series on forced lattice vibrations in which a semi‐infinite lattice of one‐dimensional particles {xn}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) = ex, the authors prove that sufficiently ample families of traveling‐wave solutions exist for all k, γk > γ > γk+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) = ex (when k = 0, take γ0 = ∞ and X0 ≡ 0).

Original languageEnglish (US)
Pages (from-to)1251-1298
Number of pages48
JournalCommunications on Pure and Applied Mathematics
Volume48
Issue number11
DOIs
StatePublished - 1995

    Fingerprint

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this