  We consider the interaction of solitary waves in a model involving the
well-known $\phi^4$ Klein-Gordon theory, but now bearing both Laplacian and
biharmonic terms with different prefactors. As a result of the competition of
the respective linear operators, we obtain three distinct cases as we vary the
model parameters. In the first the biharmonic effect dominates, yielding an
oscillatory inter-wave interaction; in the third the harmonic effect prevails
yielding exponential interactions, while we find an intriguing linearly
modulated exponential effect in the critical second case, separating the above
two regimes. For each case, we calculate the force between the kink and
antikink when initially separated with sufficient distance. Being able to write
the acceleration as a function of the separation distance, and its
corresponding ordinary differential equation, we test the corresponding
predictions, finding very good agreement, where appropriate, with the
corresponding partial differential equation results. Where the two findings
differ, we explain the source of disparities. Finally, we offer a first glimpse
of the interplay of harmonic and biharmonic effects on the results of
kink-antikink collisions and the corresponding single- and multi-bounce
windows.
