Travelling kinks in discrete ϕ4 models

In recent years, three exceptional discretizations of the ϕ4 theory have been discovered (by Speight and Ward, Bender and Tovbis, and Kevrekidis) which support translationally invariant kinks, i.e. families of stationary kinks centred at arbitrary points between the lattice sites. It has been suggested that the translationally invariant stationary kinks may persist as sliding kinks, i.e. discrete kinks travelling at nonzero velocities without experiencing any radiation damping. The purpose of this study is to check whether this is indeed the case. By computing the Stokes constants in beyond-all-order asymptotic expansions, we prove that the three exceptional discretizations do not support sliding kinks for most values of the velocity—just like the standard, one-site discretization. There are, however, isolated values of velocity for which radiationless kink propagation becomes possible. There is one such value for the discretization of Speight and Ward and three sliding velocities for the model of Kevrekidis.