  We study kink-antikink collisions in a model which interpolates smoothly
between the completely integrable sine-Gordon theory, the $\phi^4$ model, and a
$\phi^6$-like model with three degenerate vacua. We find a rich variety of
behaviours, including integrability breaking, resonance windows with
increasingly irregular patterns, and new types of windows near the
$\phi^6$-like regime. False vacua, extra kink modes and kink fragmentation play
important roles in the explanations of these phenomena. Our numerical studies
are backed up by detailed analytical considerations.
