2022/11/15 by Lizarraga, Ian, Marangell, Robert
#35G20 #37C75 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.07824
We determine the nonlinear stability of shock-fronted travelling waves arising in a reaction-nonlinear diffusion PDE, subject to a fourth-order spatial derivative term multiplied by a small parameter ε that models \it nonlocal regularization. Motivated by the authors' recent stability analysis of shock-fronted travelling waves under viscous relaxation, our numerical analysis is guided by the observation that there is a fast-slow decomposition of the associated eigenvalue problem for the linearised operator. In particular, we observe an astonishing reduction of the complex four-dimensional eigenvalue problem into a \it real one-dimensional problem defined along the slow manifolds; i.e. slow eigenvalues defined near the tails of the shock-fronted wave for ε = 0 govern the point spectrum of the linearised operator when 0 < ε ≪ 1.