  We study the dynamics of coherent waves in nonlinear honeycomb lattices and
show that nonlinearity breaks down the Dirac dynamics. As an example, we
demonstrate that even a weak nonlinearity has major qualitative effects one of
the hallmarks of honeycomb lattices: conical diffraction. Under linear
conditions, a circular input wave-packet associated with the Dirac point
evolves into a ring, but even a weak nonlinearity alters the evolution such
that the emerging beam possesses triangular symmetry, and populates Bloch modes
outside of the Dirac cone region. Our results are presented in the context of
optics, but we propose a scheme to observe equivalent phenomena in
Bose-Einstein condensates.
