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

Michael Herrmann, Karsten Matthies, Harmut Schwetlick, Johannes Zimmer

Research output: Contribution to journalArticle

8 Citations (Scopus)
156 Downloads (Pure)

Abstract

Although phase transition waves in atomic chains with double-well potential play a fundamental role in materials science, 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 localized with respect to the strain variable. As a standard Lyapunov--Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterize the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive on 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 relations.
Original languageEnglish
Pages (from-to)2625-2645
Number of pages21
JournalSIAM Journal on Mathematical Analysis (SIMA)
Volume45
Issue number5
DOIs
Publication statusPublished - 3 Sep 2013

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