  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.
