Admissible speeds in spatially periodic bistable reaction-diffusion equations

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