Riemann solvers and undercompressive shocks of convex FPU chains

Riemann solvers and undercompressive shocks of convex FPU chains

We consider FPU-type atomic chains with general convex potentials. The naive continuum limit in the hyperbolic space-time scaling is the p-system of mass and momentum conservation. We systematically compare Riemann solutions to the p-system with numerical solutions to discrete Riemann problems in FPU chains, and argue that the latter can be described by modified p-system Riemann solvers. We all...

Unimodal wave trains and solitons in convex FPU chains

We consider atomic chains with nearest neighbour interactions and study periodic and homoclinic travelling waves which are called wave trains and solitons, respectively. Our main result is a new existence proof which relies on the constrained maximisation of the potential energy and exploits the invariance properties of an improvement operator. The approach is restricted to convex interaction p...

Undercompressive Shocks and Riemann Problems for Scalar Conservation Laws with Nonconvex Fluxes

The Riemann initial value problem is studied for scalar conservation laws whose uxes have a single innection point. For a regularization consisting of balanced diiusive and dispersive terms, the traveling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for...

Heteroclinic Travelling Waves in Convex FPU-Type Chains

We consider infinite FPU-type atomic chains with general convex potentials and study the existence of monotone fronts that are heteroclinic travelling waves connecting constant asymptotic states. Iooss showed that small amplitude fronts bifurcate from convex-concave turning points of the force. In this paper we prove that fronts exist for any asymptotic states that satisfy a certain algebraic c...

Nonlinear Dynamics of Two-Dimensional Undercompressive Shocks