Nonlinear stability of shock-fronted travelling waves in reaction-nonlinear diffusion equations

Abstract

Reaction-nonlinear diffusion PDEs can be derived as continuum limits of stochastic models for biological and ecological invasion. We numerically investigate the nonlinear stability of shock-fronted travelling waves arising in these RND PDEs, in the presence of a fourth-order spatial derivative multiplied by a small parameter ɛ that models high-order regularization. Once we have verified sectoriality of our linear operator, our task is reduced to checking spectral stability of our family of travelling waves. Motivated by the authors’ recent stability analysis of shock-fronted travelling waves under viscous relaxation, our numerical analysis suggests that near the singular limit, the associated eigenvalue problem for the linearized operator admits a fast–slow decomposition similar to that constructed by Alexander, Gardner, and Jones in the early 90s. In particular, our numerical results suggest a reduction of the complex four-dimensional eigenvalue problem into a real one-dimensional problem defined along the slow manifolds; i.e. slow eigenvalues defined near the tails of the shock-fronted wave for ɛ=0 govern the point spectrum of the linearized operator when 0<ɛ≪1.