  We consider chiral, generally nonlinear density waves in one dimension,
modelling the bosonized edge modes of a two-dimensional fermionic topological
insulator. Using the coincidence between bosonization and Lie-Poisson dynamics
on an affine U(1) group, we show that wave profiles which are periodic in time
produce Berry phases accumulated by the underlying fermionic field. These
phases can be evaluated in closed form for any Hamiltonian, and they serve as a
diagnostic of nonlinearity. As an explicit example, we discuss the Korteweg-de
Vries equation, viewed as a model of nonlinear quantum Hall edge modes.
