On the dynamics of nonlinear particle chains

  • Thomas Kriecherbauer (Universität München)
A3 01 (Sophus-Lie room)


In this talk we discuss the dynamics of one-dimensional

lattices tex2html_wrap_inline11 with nearest neighbor interactions,


initially at rest, and which are driven from one end by a particle tex2html_wrap_inline15


The driver, tex2html_wrap_inline15, is assumed to undergo a prescribed



where a, tex2html_wrap_inline23,

tex2html_wrap_inline25 are real constants and

tex2html_wrap_inline27 has period

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

We describe the numerically observed behaviour (shock and rarefaction

phenomena for tex2html_wrap_inline31, generation of multi-phase travelli

ng waves

for tex2html_wrap_inline33) and present corresponding analytical result


Special emphasis is given to the integrable model tex2html_wrap_inline35


(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


Anne Dornfeld

MPI for Mathematics in the Sciences Contact via Mail

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