Generalized transition waves and their properties

TL;DR

This paper introduces a broad framework for understanding wave phenomena in reaction-diffusion systems, establishing existence, properties, and classification of generalized transition waves in time-dependent settings.

Contribution

It generalizes classical wave notions to a unified setting involving hypersurfaces and proves existence, properties, and classification results for these generalized waves.

Findings

Existence of new generalized waves in reaction-diffusion equations.

Monotonicity and uniqueness properties of almost planar fronts.

Robustness of the generalized wave definition under various conditions.

Abstract

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