A simple approach to the wave uniqueness problem

TL;DR

This paper introduces a novel method for establishing the uniqueness of semi-wavefront solutions in complex reaction-diffusion equations with delay, addressing open problems and demonstrating broad applicability.

Contribution

It presents a new approach to prove semi-wavefront uniqueness in non-monotone reaction-diffusion equations with delay, including the KPP-Fisher and Mackey-Glass type equations.

Findings

Proves uniqueness of semi-wavefronts for delayed KPP-Fisher equations.

Establishes unique semi-wavefronts for Mackey-Glass type equations.

Addresses open problem on non-monotone wavefronts.

Abstract

We propose a new approach for proving uniqueness of semi-wavefronts in generally non-monotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.