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 τ₀. The driver, τ₀, is assumed to undergo a prescribed motion, 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 ε ≠ 0) and present corresponding analytical results.

Special emphasis is given to the integrable model F(x) = ε^τ (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.