Shock selection in reaction--diffusion equations with partially negative diffusivity using nonlinear regularisation

TL;DR

This paper investigates how nonlinear regularisation in reaction-diffusion equations with negative diffusivity can select shock positions, leading to continuous diffusivity shocks with predictable locations and wave speeds, confirmed by numerical and theoretical analysis.

Contribution

It introduces a nonlinear regularisation approach that uniquely determines shock positions in reaction-diffusion equations with negative diffusivity, extending previous shock selection criteria.

Findings

Nonlinear regularisation yields unique shock positions.

Shock solutions with continuous diffusivity are constructed and analyzed.

Numerical results match theoretical predictions for shock location and wave speed.

Abstract

We consider a general reaction--nonlinear-diffusion equation with a region of negative diffusivity, and show how a nonlinear regularisation selects a shock position. Negative diffusivity can model population aggregation, but leads to shock-fronted solutions for population density. In general the shock position is non-unique. Previous studies have defined shock selection criteria such as the equal area rule, and shown how these arise from specific regularisations to the reaction--diffusion equation. In this work, we show that a nonlinear regularisation leads to travelling wave solutions where the shock is selected according to a modified equal area rule. Adjusting the nonlinearity in the regularisation moves the shock location. We focus on attaining shocks that conserve diffusivity across the shock, and demonstrate that this condition yields the longest possible shock length. Using geometric singular perturbation theory, we prove the existence of shock-fronted travelling wave solutions with continuous diffusivity, show how to construct them, and demonstrate that they correspond to a unique wave speed. Numerical solutions align with theoretical predictions for shock position and wave speed, confirming that a single regularisation term can vary the shock position and attain shocks with continuous diffusivity.