Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Forced Lattice Vibrations -- A Videotext

Percy Deift, Thomas Kriecherbauer, Stephanos Venakides

TL;DR

The paper develops a comprehensive framework for understanding forced lattice vibrations in driven, semi-infinite lattices, with a focus on the Toda lattice and periodic boundary forcing. It combines a strongly nonlinear spectral approach (via evolution of Lax spectra and a continuum limit yielding an integral equation for the asymptotic spectral density J(λ)) with a boundary-matching method that reduces the problem to solving for Fourier coefficients in weighted function spaces, exploiting nonresonant and resonant modes. For general lattices, traveling wave constructions using Lyapunov-Schmidt reductions illustrate how to realize time-periodic responses, while in the Toda case the g-gap solutions provide explicit, analytically tractable traveling-wave families that generate periodic driven states; small-gap analysis connects spectral data to observable band-gap widths. The results culminate in a rigorous-like description of the long-time attractors: periodic multiphase waves behind the driving boundary, with transport of energy into the lattice, and a structured band-gap evolution that mirrors the drive frequency and amplitude. Together, these contributions offer a detailed bridge between nonlinear lattice dynamics, spectral theory, and integrable-system techniques with potential applications to nonlinear wave propagation in discrete media.

Abstract

We begin with a description of recent numerical and analytical results that are closely related to the results of this paper.

Forced Lattice Vibrations -- A Videotext

TL;DR

The paper develops a comprehensive framework for understanding forced lattice vibrations in driven, semi-infinite lattices, with a focus on the Toda lattice and periodic boundary forcing. It combines a strongly nonlinear spectral approach (via evolution of Lax spectra and a continuum limit yielding an integral equation for the asymptotic spectral density J(λ)) with a boundary-matching method that reduces the problem to solving for Fourier coefficients in weighted function spaces, exploiting nonresonant and resonant modes. For general lattices, traveling wave constructions using Lyapunov-Schmidt reductions illustrate how to realize time-periodic responses, while in the Toda case the g-gap solutions provide explicit, analytically tractable traveling-wave families that generate periodic driven states; small-gap analysis connects spectral data to observable band-gap widths. The results culminate in a rigorous-like description of the long-time attractors: periodic multiphase waves behind the driving boundary, with transport of energy into the lattice, and a structured band-gap evolution that mirrors the drive frequency and amplitude. Together, these contributions offer a detailed bridge between nonlinear lattice dynamics, spectral theory, and integrable-system techniques with potential applications to nonlinear wave propagation in discrete media.

Abstract

We begin with a description of recent numerical and analytical results that are closely related to the results of this paper.

Paper Structure

This paper contains 29 sections, 30 theorems, 370 equations, 19 figures.

Table of Contents

  1. Introduction
  2. An asymptotic calculation in the strongly nonlinear case
  3. The evolution equations
  4. The continuum limit of eigenvalue dynamics
  5. The asymptotic spectral density of $L_{N}$
  6. A boundary matching technique
  7. Introduction
  8. The equation for the Fourier coefficients
  9. Sequence spaces
  10. The general assumptions
  11. The nonlinear terms
  12. The equations for the Fourier coefficients
  13. Solving for the non resonant modes
  14. An immediate application: high frequency driving
  15. General Lattices
  16. ...and 14 more sections

Key Result

Theorem 2.4

The evolution of the $\lambda_j$'s and $f_j$'s is given by: where $\rho=2b_0^2(t) = -\sum^N_{i=1} \dot\lambda_i$. The initial values $\lambda_i(0)$ are the eigenvalues of $L_N$ at $t=0$ while the initial values $\dot\lambda_i(0)$ are given by

Figures (19)

  • Figure 1.1: The canonical homology basis of the Riemann surface
  • Figure 1.4: Motion of the first ten particles of a lattice described by the above system (1.1) -- (1.3) with $F(x) = e^x, d = 0, a=.5$, in the frame of $x_0$ (case $a < a_{{\rm crit}}$).
  • Figure 1.5: Motion of the first ten particles of a lattice described by the above system (1.1) -- (1.3) with $F(x) = e^x, d = 0, a=2$, in the frame of $x_0$ (case $a > a_{{\rm crit}}$).
  • Figure 3.6: Motion of lattices (cf (C.1) -- (C.3)) with $\varepsilon = 0.2, \gamma = 3.1$
  • Figure 3.7: Motion of lattices (cf (C.1) -- (C.3)) with $\varepsilon = 0.2, \gamma = 2.1$
  • ...and 14 more figures

Theorems & Definitions (49)

  • Theorem 2.4
  • Theorem 2.27
  • Definition 2.29
  • Theorem 2.38
  • Remark 2.46
  • Definition 3.15
  • Definition 3.18
  • Definition 3.21
  • Definition 3.22
  • Proposition 3.23
  • ...and 39 more