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

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 multi...

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Bibliographic Details
Published in:Physica. D Vol. 460; p. 134069
Main Authors: Lizarraga, Ian, Marangell, Robert
Format: Journal Article
Language:English
Published: Elsevier B.V 01.04.2024
Subjects:
Geometric singular perturbation theory
Riccati–Evans functions
Shock-fronted travelling waves
Stability theory for travelling waves
Geometric singular perturbation theory
Riccati–Evans functions
Stability theory for travelling waves
Shock-fronted travelling waves
ISSN:0167-2789, 1872-8022
Online Access:Get full text
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Summary: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. •Nonlinear stability of shock-fronted travelling waves in regularized RND PDEs.•Geometric Riccati–Evans function analysis of the spectral problem.•Fast–slow subbundle theory explains spectral stability calculations.
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2024.134069