On the dynamics of nonlinear particle chains

In this talk we discuss the dynamics of one-dimensional

lattices with nearest neighbor interactions,

initially at rest, and which are driven from one end by a particle

="krie/img3.gif">.

The driver, , is assumed to undergo a prescribed

motion,

where a, ,

are real constants and

has period

LT="tex2html_wrap_inline29" SRC="krie/img8.gif">.

We describe the numerically observed behaviour (shock and rarefaction

phenomena for , generation of multi-phase travelli

ng waves

for ) and present corresponding analytical result

s.

Special emphasis is given to the integrable model

>

(Toda lattice). In this particular case one can rigorously derive

the long-time asymptotics for a large class of initial value problems

using the Inverse Scattering Transform (IST) method. Hereby

we formulate the IST as a matrix-valued Riemann-Hilbert problem.

Such Riemann-Hilbert problems have recently been used to prove

asymptotic results in a variety of different fields, such as

integrable systems, statistical mechanics, combinatorics and orthogonal

polynomials.