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arXiv · 1206.0268

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

Michael Herrmann·Karsten Matthies·Hartmut Schwetlick·Johannes Zimmer
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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.

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Michael Herrmann, Karsten Matthies, Hartmut Schwetlick, Johannes Zimmer. 2012-06-01. Subsonic phase transition waves in bistable lattice models with small spinodal region. https://doi.org/10.1137/120877878

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