Uniqueness and stability properties of monostable pulsating fronts

International audienceIn this paper, we prove the uniqueness, up to shifts, of pulsating traveling fronts for reaction-diffusion equations in periodic media with Kolmogorov-Petrovsky-Piskunov type nonlinearities. These results provide in particular a complete classification of all KPP pulsating fronts. Furthermore, in the more general case of monostable nonlineari-ties, we also derive several global stability properties and convergence to the pulsating fronts for the solutions of the Cauchy problem with front-like initial data. In particular, we prove the stability of KPP pulsating fronts with minimal speed, which is a new result even in the case when the medium is invariant in the direction of propagation