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Reaction-nonlinear diffusion (RND) partial differential equations are afruitful playground to model the formation of sharp travelling fronts, afundamental pattern in nature. In this work, we demonstrate the utility andscope of regularisation as a technique to investigate shock-fronted solutionsof RND PDEs, using geometric singular perturbation theory (GSPT) as themathematical framework. In particular, we show that composite regularisationscan be used to construct families of monotone shock-fronted travelling wavessweeping out distinct generalised area rules, which interpolate between theequal area and extremal area (i.e. algebraic decay) rules that are well-knownin the shockwave literature. We further demonstrate that our RND PDE supportsother kinds of shock-fronted solutions, namely, nonmonotone shockwaves as wellas shockwaves containing slow tails in the aggregation (negative diffusion)regime. Our analysis blends Melnikov methods -- in both smooth andpiecewise-smooth settings -- with GSPT techniques applied to the PDE overdistinct spatiotemporal scales.  We also consider the spectral stability of these new interpolated shockwaves.Using techniques from geometric spectral stability theory, we determine thatour RND PDE admits spectrally stable shock-fronted travelling waves. Themultiple-scale nature of the regularised RND PDE continues to play an importantrole in the analysis of the spatial eigenvalue problem.

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