  This paper is devoted to the study of some qualitative and quantitative
aspects of nonlinear propagation phenomena in diffusive media. More precisely,
we consider the case a reaction-diffusion equation in a periodic medium with
ignition-type nonlinearity, the heterogeneity being on the nonlinearity, the
operator and the domain. Contrary to previous works, we study the asymptotic
spreading properties of the solutions of the Cauchy problem with general
initial conditions which satisfy very mild assumptions at infinity. We
introduce several concepts generalizing the notion of spreading speed and we
give a complete characterization of it when the initial condition is
asymptotically oscillatory at infinity. Furthermore we construct, even in the
homogeneous one-dimensional case, a class of initial conditions for which
highly nontrivial dynamics can be exhibited.
