A geometric singular perturbation analysis of generalised shock selection rules in reaction-nonlinear diffusion models

Abstract

Reaction-nonlinear diffusion (RND) partial differential equations are a fruitful playground to model the formation of sharp travelling fronts, a fundamental pattern in nature. In this work, we demonstrate the utility and scope of regularisation as a technique to investigate shock-fronted solutions of RND PDEs, using geometric singular perturbation theory (GSPT) as the mathematical framework. In particular, we show that composite regularisations can be used to construct families of monotone shock-fronted travelling waves sweeping out distinct generalised area rules, which interpolate between the equal area and extremal area (i.e. algebraic decay) rules that are well-known in the shockwave literature. We further demonstrate that our RND PDE supports other kinds of shock-fronted solutions, namely, nonmonotone shockwaves as well as shockwaves containing slow tails in the aggregation (negative diffusion) regime. Our analysis blends Melnikov methods -- in both smooth and piecewise-smooth settings -- with GSPT techniques applied to the PDE over distinct spatiotemporal scales. We also consider the spectral stability of these new interpolated shockwaves. Using techniques from geometric spectral stability theory, we determine that our RND PDE admits spectrally stable shock-fronted travelling waves. The multiple-scale nature of the regularised RND PDE continues to play an important role in the analysis of the spatial eigenvalue problem.