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We explore a variant of the $\phi^6$ model originally proposed in Phys.\Rev.\ D {\bf 12}, 1606 (1975) as a prototypical, so-called, "bag" model inwhich domain walls play the role of quarks within hadrons. We examine thesteady state of the model, namely an apparent bound state of two kinkstructures. We explore its linearization, and we find that, as a function of aparameter controlling the curvature of the potential, an {\it effectivelyarbitrary} number of internal modes may arise in the point spectrum of thelinearization about the domain wall profile. We explore some of the keycharacteristics of kink-antikink collisions, such as the critical velocity andthe multi-bounce windows, and how they depend on the principal parameter of themodel. We find that the critical velocity exhibits a non-monotonic dependenceon the parameter controlling the curvature of the potential. For themulti-bounce windows, we find that their range and complexity decrease as therelevant parameter decreases (and as the number of internal modes in the modelincreases). We use a modified collective coordinates method [in the spirit ofrecent works such as {Phys.\ Rev.\ D} {\bf 94}, 085008 (2016)] in order tocapture the relevant phenomenology in a semi-analytical manner.

Kink Dynamics in a Parametric $\phi^6$ System: A Model With Controllably Many Internal Modes