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

TL;DR

This paper investigates the mathematical properties of phase transition waves in atomic chains with double-well potentials, demonstrating the persistence and uniqueness of such waves under small localized perturbations of a bi-quadratic potential.

Contribution

It extends the understanding of phase transition waves by proving their persistence and uniqueness for perturbed bi-quadratic potentials using a novel fixed point approach.

Findings

Persistence of three-parameter family of waves under small perturbations

Characterization of wave perturbation as a contractive fixed point

Discussion of the kinetic relation for phase transition waves

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.