  Spatially periodic reaction-diffusion equations typically admit pulsating
waves which describe the transition from one steady state to another. Due to
the heterogeneity, in general such an equation is not invariant by rotation and
therefore the speed of the pulsating wave may a priori depend on its direction.
However, little is actually known in the literature about whether it truly
does: surprisingly, it is even known in the one-dimensional monostable
Fisher-KPP case that the speed is the same in the opposite directions despite
the lack of symmetry. Here we investigate this issue in the bistable case and
show that the set of admissible speeds is actually rather large, which means
that the shape of propagation may indeed be asymmetrical. More precisely, we
show in any spatial dimension that one can choose an arbitrary large number of
directions , and find a spatially periodic bistable type equation to achieve
any combination of speeds in those directions, provided those speeds have the
same sign. In particular, in spatial dimension 1 and unlike the Fisher-KPP
case, any pair of (either nonnegative or nonpositive) rightward and leftward
wave speeds is admissible. We also show that these variations in the speeds of
bistable pulsating waves lead to strongly asymmetrical situations in the
multistable equations.
