On the dynamics of nonlinear particle chains

In this talk we discuss the dynamics of one-dimensional lattices $(x_n)_{n\geq 1}$ with nearest neighbor interactions, $$\overline{x}_n = F(x_{n-1} - x_n ) - F(x_n - x_{n+1}) , \ \ n\geq 1$$ initially at rest, and which are driven from one end by a particle $x_0$. The driver, $x_0$, is assumed to undergo a prescribed motion, $$x_0 (t) = at + \epsilon h (\gamma t)$ where $a , \epsilon , \gamma$$ are real constants and $h(\cdot)$ has period $2\pi$. We describe the numerically observed behaviour (shock and rarefaction phenomena for $\epsilon = 0$, generation of multi-phase travelling waves for $\epsilon \neq 0$) and present corresponding analytical results. Special emphasis is given to the integrable model $F(x) = e^x$ (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.