Rapid travelling waves in the nonlocal Fisher equation connect two unstable states

TL;DR

This paper proves the existence of rapid traveling wave solutions connecting two unstable states in the nonlocal Fisher equation, including cases with Turing instability and fat-tailed kernels, without using maximum principles.

Contribution

It establishes the existence of rapid traveling waves connecting unstable states for any kernel and slope, extending previous results to more general kernels and conditions.

Findings

Existence of rapid traveling waves connecting unstable states.

Applicability to kernels with fat tails.

No maximum principle used in the proof.

Abstract

In this note, we give a positive answer to a question addressed in \cite{Nad-Per-Tan}. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are "rapid") of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with {\it fat tails}.