Quantized nonlinear kink movement through topological boundary state instabilities

Thouless pumping is a paradigmatic example of topologically protected, directed transport in linear systems. Recent extensions to nonlinear pumps often overlook the need to reassess the conventional framework of linear topology. In this work, we study a nonlinear dimer-chain model that exhibits quantized transport of kinks under a periodic modulation of a pumping parameter. Crucially, linear excitations in the system map to a Rice-Mele model and display topological boundary modes localized at these kinks. Using methods from nonlinear dynamics, we show that instabilities in these boundary modes are the driving mechanism behind the observed kink motion. While the transport resembles that of a linear Thouless pump, it cannot be fully captured by conventional topological indices. Instead, the behavior is more akin to a topological ratchet: robust, directional, and reproducible, yet fundamentally nonlinear. Furthermore, by introducing multiple pumping parameters, we demonstrate fine control over multiple kink trajectories, as well as soliton motion, suggesting applications in information transport. Our results unify concepts from linear topology and nonlinear dynamics to establish a framework for quantized transport in nonlinear media.