Nonlinear stability of shock-fronted travelling waves under nonlocal regularization

TL;DR

This paper investigates the nonlinear stability of shock-fronted travelling waves in a reaction-nonlinear diffusion PDE with nonlocal regularization, revealing a reduction of the eigenvalue problem to a simpler form that governs stability.

Contribution

The authors analyze the stability of shock-fronted waves under nonlocal regularization, demonstrating a reduction of the eigenvalue problem to a one-dimensional real problem near the tails.

Findings

Eigenvalue problem reduces from four-dimensional to one-dimensional.

Slow eigenvalues near the tails determine stability for small regularization parameter.

Numerical analysis confirms the theoretical reduction and stability criteria.

Abstract

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 $\varepsilon$ 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 $\varepsilon = 0$ govern the point spectrum of the linearised operator when $0 < \varepsilon \ll 1$.