To show that steadily propagating nonlinear waves in active matter can be driven internally, we develop a prototypical model of a topological kink moving with a constant supersonic speed. We use a model of a bi-stable mass-spring (Fermi–Pasta–Ulam) chain capable of generating active stress. In contrast to subsonic kinks in passive bi-stable chains that are necessarily dissipative, the obtained supersonic solutions are purely anti-dissipative. Our numerical experiments point towards the stability of the obtained kink-type solutions and the possibility of propagating kink-anti-kink bundles reminiscent of solitons. We show that even the simplest quasi-continuum approximation of the discrete model captures the most important features of the predicted active phenomena. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.