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

Abstract

Solutions to reaction–nonlinear-diffusion (RND) equations with a region of negative diffusivity exhibit shocks. In general, the position of these shocks can vary, necessitating selection criteria to determine a unique shock. Previous studies have defined conditions for shock selection. A common choice is the equal area rule, which corresponds to a fourth-order non-local regularisation to the RND equation. Bradshaw-Hajek et al. (2024) showed that combining non-local and viscous regularisations can yield a continuum of possible shocks. In this work, we demonstrate how to achieve a continuum of shocks using a single nonlinear regularisation term. With one nonlinear regularisation, shock selection obeys a modified equal area rule, where adjusting the nonlinearity in the regularisation moves the shock. To demonstrate the technique, we attain solutions with conserved diffusivity across the shock, which yield the longest shock length possible. Using geometric singular perturbation theory, we prove the existence of travelling waves with continuous diffusivity shocks. Numerical solutions align with theoretical predictions for shock position and wave speed.