  The quantum-mechanical counterpart of a classical random walk offers a rich
dynamics that has recently been shown to include topologically protected bound
states (zero-modes) at boundaries or domain walls. Here we show that a
topological zero-mode may acquire a dynamical role in the presence of
nonlinearities. We consider a one-dimensional discrete-time quantum walk that
combines zero-modes with a particle-conserving nonlinear relaxation mechanism.
The presence of both particle-hole and chiral symmetry converts two zero-modes
of opposite chirality into an attractor-repeller pair of the nonlinear
dynamics. This makes it possible to steer the walker towards a domain wall and
trap it there.
