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

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Travelling kinks in discrete phi-4 models

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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