# Pulsating solutions for multidimensional bistable and multistable equations

## TL;DR

This paper investigates the existence and structure of pulsating traveling front solutions in multidimensional reaction-diffusion equations with spatial heterogeneity, revealing complex phenomena like propagating terraces whose configurations depend on propagation direction.

## Contribution

It introduces the concept of propagating terraces in multistable equations and shows their dependence on propagation direction, extending understanding of wave solutions in heterogeneous media.

## Key findings

- Existence of pulsating traveling fronts in multidimensional heterogeneous media.
- Identification of propagating terraces as a key solution structure.
- Dependence of terrace shape on propagation direction.

## Abstract

We devote this paper to the issue of existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.