O.F. Oxtoby, D.E. Pelinovsky, and I.V. Barashenkov,

Travelling kinks in discrete phi-4 models

Nonlinearity 19, 217-235 (2006)

Abstract:
In recent years, three exceptional discretizations of the phi-4 model have been discovered which support translationally invariant kinks, i.e., families of stationary kinks centered at arbitrary points between the lattice sites. We address the question of the persistence of these kinks for nonzero velocity. By computing the Stokes constants in beyond-all-order asymptotic expansions, we prove that these three discretizations fail to support radiationless travelling kinks for almost all nonzero values of the kink velocity. We show, however, that there exists a travelling kink for one particular value of velocity when the discretization is as prescribed in (Speight, 1997), and for three values of the velocity when the discretization from (Kevrekidis, 2003) is used.

Keywords:
DISCRETE PHI-4 EQUATION, STATIONARY KINKS, TRAVELING KINKS, EXISTENCE PROBLEM, BEYOND-ALL-ORDER ASYMPTOTIC EXPANSIONS, STOKES CONSTANTS, NUMERICAL METHODS