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.
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