  In many contexts, the interaction between particles gives rise to emergent
and perhaps unanticipated physical phenomena. An example is the fractional
quantum Hall effect, where interaction between electrons gives rise to
fractionally quantized Hall conductance. In photonic systems, the nonlinear
response of an ambient medium acts to mediate interaction between photons; in
the mean-field limit these dynamics are described by the nonlinear
Schr\"odinger (also called Gross-Pitaevskii) equation. Recently, it was shown
that at weak nonlinearity, soliton motion in nonlinear Thouless pumps (a
dimensionally reduced implementation of a Chern insulator) could be quantized
to the Chern number of the band from which the soliton bifurcates. Here, we
show theoretically and experimentally using arrays of coupled optical
waveguides that sufficiently strong nonlinearity acts to fractionally quantize
the motion of solitons. Specifically, we find that the soliton returns to
itself after multiple cycles of the Thouless pump - but displaced by an integer
number of unit cells - leading to a rich fractional plateaux structure
describing soliton motion. Our results demonstrate a perhaps surprising example
of the behavior of non-trivial topological systems in the presence of
interactions.
