Uniqueness of solitary waves in the high-energy limit of FPU-type chains

Recent asymptotic results by the authors provided detailed information on the shape of solitary high-energy travelling waves in FPU atomic chains. In this note we use and extend the methods to understand the linearisation of the travelling wave equation. We show that there are not any other zero eigenvalues than those created by the translation symmetry and this implies a local uniqueness result. The key argument in our asymptotic analysis is to replace the linear advance-delay-differential equation for the eigenfunctions by an approximate ODE

Action minimizing fronts in general FPU-type chains

We study atomic chains with nonlinear nearest neighbour interactions and prove the existence of fronts (heteroclinic travelling waves with constant asymptotic states). Generalizing recent results of Herrmann and Rademacher we allow for non-convex interaction potentials and find fronts with non-monotone profile. These fronts minimize an action integral and can only exists if the asymptotic states fulfil the macroscopic constraints and if the interaction potential satisfies a geometric graph condition. Finally, we illustrate our findings by numerical simulations.Comment: 19 pages, several figure

Asymptotic formulas for solitary waves in the high-energy limit of FPU-type chains

It is well established that the solitary waves of FPU-type chains converge in the high-energy limit to traveling waves of the hard-sphere model. In this paper we establish improved asymptotic expressions for the wave profiles as well as an explicit formula for the wave speed. The key step in our approach is the derivation of an asymptotic ODE for the appropriately rescaled strain profile.Comment: revised version with corrected typos; 25 pages, several figure

Subsonic phase transition waves in bistable lattice models with small spinodal region

Phase transitions waves in atomic chains with double-well potential play a fundamental role in materials science, but very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewise-quadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the corresponding three-parameter family of waves persists as long as the perturbation is small and localised with respect to the strain variable. As a standard Lyapunov-Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterise the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive in a small ball in a suitable function space. Moreover, we derive a uniqueness result for phase transition waves with certain properties and discuss the kinetic relation.Comment: revised version with extended introduction, improved perturbation method, and novel uniqueness result; 20 pages, 5 figure

On selection criteria for problems with moving inhomogeneities

We study mechanical problems with multiple solutions and introduce a thermodynamic framework to formulate two different selection criteria in terms of macroscopic energy productions and fluxes. Studying simple examples for lattice motion we then compare the implications for both resting and moving inhomogeneities.Comment: revised version contains new introduction, numerical simulations of Riemann problems, and a more detailed discussion of the causality principle; 18 pages, several figure

Uniqueness of solitary waves in the high-energy limit of FPU-type chains

Recent asymptotic results by the authors provided detailed information on the shape of solitary high-energy travelling waves in FPU atomic chains. In this note we use and extend the methods to understand the linearisation of the travelling wave equation. We show that there are not any other zero eigenvalues than those created by the translation symmetry and this implies a local uniqueness result. The key argument in our asymptotic analysis is to replace the linear advance-delay-differential equation for the eigenfunctions by an approximate ODE