  We generalize the concept of optical Galton board (OGB), first proposed by
Bouwmeester et al. {[}Phys. Rev. A \textbf{61}, 013410 (2000)], by introducing
the possibility of nonlinear self--phase modulation on the wavefunction during
the walker evolution. If the original Galton board illustrates classical
diffusion, the OGB, which can be understood as a grid of Landau--Zener
crossings, illustrates the influence of interference on diffusion, and is
closely connected with the quantum walk. Our nonlinear generalization of the
OGB shows new phenomena, the most striking of which is the formation of
non-dispersive pulses in the field distribution (soliton--like structures).
These exhibit a variety of dynamical behaviors, including ballistic motion,
dynamical localization, non--elastic collisions and chaotic behavior, in the
sense that the dynamics is very sensitive to the nonlinearity strength.
