Quasi-continuum approximations for travelling kinks in diatomic lattices

We extend the quasi-continuum method of approximation to the case of a diatomic lattice. This is illustrated by a lattice in which both small and large atoms interact with first- and second-nearest neighbours. We show that highly accurate quasi-continuum techniques may be generalized to determine the shape of nontopological kinks. Many previous analyses of such systems have found travelling wave solutions only for one particular speed using a second-order continuum theory; we show (i) how wave shapes can be found for arbitrary speeds, and (ii) how solution profiles can be calculated from fourth-order partial differential equations which approximate the lattice. Alongside the theoretical analysis, we also present numerical simulations of the lattice which demonstrate propagation of the predicted waves through the lattice. We show that the particular speed for which solutions have been found in previous studies is a special speed for waves in the lattice, but waves can travel for long periods of time at faster or slower speeds, whilst slowly relaxing to this critical speed.