Generalized transition waves and their properties

In this paper, we generalize the usual notions of waves, fronts and propagation speeds in a very general setting. These new notions, which cover all classical situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. We prove the existence of new such waves for some time-dependent reaction-diffusion equations, as well as general intrinsic properties, some monotonicity properties and uniqueness results for almost planar fronts. The classification results, which are obtained under appropriate assumptions, show the robustness of our general definition