  We study collisions of coherent structures in higher-order field-theoretic
models, such as the $\phi^8$, $\phi^{10}$ and $\phi^{12}$ ones. The main
distinguishing feature, of the example models considered herein, is that the
collision arises due to the long-range interacting algebraic tails of these
solitary waves. We extend the approach to suitably initialize the relevant
kinks, in the additional presence of finite initial speed, in order to minimize
the dispersive wave radiation potentially created by their slow spatial decay.
We find that, when suitably initialized, these models still feature the
multi-bounce resonance windows earlier found in models in which the kinks bear
exponential tails, such as the $\phi^4$ and $\phi^6$ field theories among
others. Also present is the self-similar structure of the associated windows
with three- and more-bounce windows at the edges of two- and lower-bounce ones.
Moreover, phenomenological, but highly accurate (and predictive) scaling
relations are derived for the dependence of the time between consecutive
collisions and, e.g., the difference in kinetic energy between the incoming one
and the critical one for one-bounces. Such scalings are traced extensively over
two-bounce collision windows throughout the three models, hinting at the
possibility of an analytical theory in this direction.
