  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.
