120 research outputs found
The Fermi-Pasta-Ulam problem: 50 years of progress
A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with
its suggested resolutions and its relation to other physical problems. We focus
on the ideas and concepts that have become the core of modern nonlinear
mechanics, in their historical perspective. Starting from the first numerical
results of FPU, both theoretical and numerical findings are discussed in close
connection with the problems of ergodicity, integrability, chaos and stability
of motion. New directions related to the Bose-Einstein condensation and quantum
systems of interacting Bose-particles are also considered.Comment: 48 pages, no figures, corrected and accepted for publicatio
Cnoidal Waves on Fermi-Pasta-Ulam Lattices
We study a chain of infinitely many particles coupled by nonlinear springs,
obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) -
V'(q_n-q_{n-1})] with generic nearest-neighbour potential . We show that
this chain carries exact spatially periodic travelling waves whose profile is
asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The
discrete waves have three interesting features: (1) being exact travelling
waves they keep their shape for infinite time, rather than just up to a
timescale of order wavelength suggested by formal asymptotic analysis,
(2) unlike solitary waves they carry a nonzero amount of energy per particle,
(3) analogous behaviour of their KdV continuum counterparts suggests long-time
stability properties under nonlinear interaction with each other. Connections
with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an
adaptation of the renormalization approach of Friesecke and Pego (1999) to a
periodic setting and the spectral theory of the periodic Schr\"odinger operator
with KdV cnoidal wave potential.Comment: 25 pages, 3 figure
KdV-like solitary waves in two-dimensional FPU-lattices
We prove the existence of solitary waves in the KdV limit of two-dimensional
FPU-type lattices using asymptotic analysis of nonlinear and singularly
perturbed integral equations. In particular, we generalize the existing results
by Friesecke and Matthies since we allow for arbitrary propagation directions
and non-unidirectional wave profiles.Comment: revised version with several changes in the presentation of the
technical details; 25 pages, 15 figure
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