Talk

On the dynamics of nonlinear particle chains

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

Abstract

In this talk we discuss the dynamics of one-dimensional lattices (xn)n1 with nearest neighbor interactions, xn=F(xn1xn)F(xnxn+1),  n1 initially at rest, and which are driven from one end by a particle x0. The driver, x0, is assumed to undergo a prescribed motion, x0(t)=at+ϵh(γt)$where$a,ϵ,γ are real constants and h() has period 2π. We describe the numerically observed behaviour (shock and rarefaction phenomena for ϵ=0, generation of multi-phase travelling waves for ϵ0) and present corresponding analytical results. Special emphasis is given to the integrable model F(x)=ex (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.

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