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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 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.