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.

revised version with extended introduction, improved perturbation method, and novel uniqueness result; 20 pages, 5 figures

References in corpus (2)

• Action minimizing fronts in general FPU-type chains
• On selection criteria for problems with moving inhomogeneities

Cited by in corpus (9)

• Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio
• KdV waves in atomic chains with nonlocal interactions
• Micropteron traveling waves in diatomic Fermi-Pasta-Ulam-Tsingou lattices under the equal mass limit
• High-energy waves in superpolynomial FPU-type chains
• Transition fronts and their universality classes
• Solitary waves in atomic chains and peridynamical media
• Fronts in dissipative Fermi-Pasta-Ulam-Tsingou chains
• Nonlinear and Nonlocal Eigenvalue Problems
• Ground waves in atomic chains with bi-monomial double-well potential