  We study general features of the excitation spectrum of a system of
one-dimensional chiral spinless fermions with short-range interactions. We show
that the nature of the elementary excitations of such a system depends strongly
on the nonlinearity of the underlying dispersion of the fermions. In the case
of quadratic nonlinearity, the low-momentum excitations are essentially
fermionic quasiparticles and quasiholes, whereas the high-momentum ones are
classical harmonic waves and solitons. In the case of cubic nonlinearity, the
nature of the elementary excitations does not depend on momentum and is
determined by the strength of the interactions. At a certain critical value of
the interaction strength the excitation spectrum changes qualitatively,
pointing to a dynamic phase transition in the system.
