  The nonlinear Schrodinger equation supports solitons -- self-interacting,
localized states that behave as nearly independent objects. We exhibit solitons
with self-induced nonreciprocal dynamics in a discrete nonlinear Schrodinger
equation. This nonreciprocal behavior, dependent on soliton power, arises from
the interplay between linear and nonlinear terms in the equations of motion.
Initially stable at high power, solitons exhibit nonreciprocal instabilities as
power decreases, leading to unidirectional acceleration and amplification. This
behavior is topologically protected by winding numbers on the solitons'
mean-field Hamiltonian and their stability matrix, linking nonlinear dynamics
and point gap topology in non-Hermitian Hamiltonians.
