The Evolution of Travelling Waves in a KPP Reaction-Diffusion Model with cut-off Reaction Rate. II. Evolution of Travelling Waves

TL;DR

This paper analyzes the long-term behavior of travelling waves in a KPP reaction-diffusion model with a discontinuous reaction rate cut-off, using asymptotic methods and numerical validation to determine wave speed corrections and convergence rates.

Contribution

It introduces a detailed asymptotic analysis of travelling wave solutions with reaction cut-offs, providing explicit corrections to wave speed and convergence rates.

Findings

Asymptotic form of solutions with reaction cut-off derived

Wave speed correction quantified through asymptotic analysis

Numerical results confirm theoretical predictions

Abstract

In Part II of this series of papers, we consider an initial-boundary value problem for the Kolmogorov--Petrovskii--Piscounov (KPP) type equation with a discontinuous cut-off in the reaction function at concentration $u=u_c$. For fixed cut-off value $u_c \in (0,1)$, we apply the method of matched asymptotic coordinate expansions to obtain the complete large-time asymptotic form of the solution which exhibits the formation of a permanent form travelling wave structure. In particular, this approach allows the correction to the wave speed and the rate of convergence of the solution onto the permanent form travelling wave to be determined via a detailed analysis of the asymptotic structures in small-time and, subsequently, in large-space. The asymptotic results are confirmed against numerical results obtained for the particular case of a cut-off Fisher reaction function.