Subsonic phase transition waves in bistable lattice models with small spinodal region
Abstract
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.
Keywords
Cite
@article{arxiv.1206.0268,
title = {Subsonic phase transition waves in bistable lattice models with small spinodal region},
author = {Michael Herrmann and Karsten Matthies and Hartmut Schwetlick and Johannes Zimmer},
journal= {arXiv preprint arXiv:1206.0268},
year = {2015}
}
Comments
revised version with extended introduction, improved perturbation method, and novel uniqueness result; 20 pages, 5 figures