  We investigate the generic transport in a one-dimensional strongly correlated
fermionic chain beyond linear response. Starting from a Gaussian wave packet
with positive momentum on top of the ground state, we find that the numerical
time evolution splits the signal into at least three distinct fractional
charges moving with different velocities. A fractional left-moving charge is
expected from conventional Luttinger liquid theory, but for the prediction of
the two separate right-moving packets the nonlinearity of the dispersion must
also be taken into account. This out-of-equilibrium protocol therefore allows a
direct measurement of nonlinear interaction parameters, which also govern
threshold singularities of dynamic response functions. The nonlinear Luttinger
Liquid theory also predicts the correct dynamics at low energies, where it
agrees with the conventional Luttinger liquid. Moreover, at high energies, the
wave packet dynamics reveals signatures of composite excitations containing
two-particle bound states. Our results uncover a simple strategy to probe the
nonlinear regime in time-resolved experiments in quantum wires and
ultracold-atom platforms.
