A Geometric Singular Perturbation Analysis of Shock Selection Rules in Composite Regularized Reaction-Nonlinear Diffusion Models

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2099-2137, September 2024.
Abstract.Reaction-nonlinear diffusion partial differential equations (RND PDEs) have recently been developed as a powerful and flexible modeling tool in order to investigate the emergence of steep fronts in biological and ecological contexts. In this work, we demonstrate the utility and scope of regularization as a technique to investigate the existence and uniqueness of steep-fronted traveling wave solutions in RND PDE models with forward-backward-forward diffusion. In a recent work (see [Y. Li et al., Phys. D, 423 (2021), 132916]), geometric singular perturbation theory (GSPT) was introduced as a framework to analyze these regularized RND PDEs. Using the GSPT toolbox, different regularizations were shown to give rise to distinct families of monotone steep-fronted traveling waves which limit to their shock-fronted singular counterparts, obeying either the equal area or extremal area (i.e., algebraic decay) rules that are well known in the shockwave literature. In this work, we extend those earlier results by showing that composite regularizations can be used to construct families of monotone shock-fronted traveling waves sweeping out distinct generalized area rules, which smoothly interpolate between these two extremal rules for shock selection. Our analysis blends Melnikov methods—including a new variant of the method which can be applied to autonomous piecewise-smooth systems—with GSPT techniques applied to the traveling wave problem of the regularized RND model over distinct spatiotemporal scales. We further demonstrate using numerical continuation that our composite model supports more exotic shock-fronted solutions, namely, nonmonotone shock-fronted waves as well as shock-fronted waves containing slow tails in the aggregation (backward diffusion) regime. We complement these existence results with a numerical spectral stability analysis of some of these new “interpolated” steep-fronted waves. Using techniques from geometric spectral stability theory, our numerical results suggest that the monotone families remain spectrally stable in the “interpolation” regime, which extends recent stability results by some of the authors in [I. Lizarraga and R. Marangell, Phys. D, 460 (2024), 134069], [I. Lizarraga and R. Marangell, J. Nonlinear Sci., 33 (2023), 82]. The multiple-scale nature of the composite regularized RND PDE model continues to play an important role in the numerical analysis of the spatial eigenvalue problem.