Analytic shock-fronted solutions to a reaction-diffusion equation with negative diffusivity

TL;DR

This paper derives analytic solutions for a reaction-diffusion equation with negative diffusivity, revealing complex wave behaviors, stability properties, and novel shock conditions, advancing understanding of aggregation phenomena in such systems.

Contribution

It introduces a new analytic approach to solving nonlinear RDEs with negative diffusivity, including shock solutions and stability analysis, which were not previously available.

Findings

Constructed explicit receding and colliding wave solutions.

Proved spectral stability of certain traveling waves.

Developed a new shock condition generalizing the equal-area rule.

Abstract

Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(\vec{x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u,$ which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behaviour in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity $D(u) = (u - a)(u - b)$ that is negative for $u\in(a,b)$. We use a nonclassical symmetry to construct analytic receding time-dependent, colliding wave, and receding travelling wave solutions. These solutions are multi-valued, and we convert them to single-valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan-like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the $u = 0$ and $u = 1$ constant solutions, and prove for certain $a$ and $b$ that receding travelling waves are spectrally stable. Additionally, we introduce a new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well-known equal-area rule, but for non-symmetric diffusivity it results in a different shock position.