Pulse replication and accumulation of eigenvalues

TL;DR

This paper studies the spectral properties of traveling pulses in slow-fast PDE systems, revealing eigenvalue accumulation phenomena linked to canard-like transitions and the absolute spectrum, with implications for pulse stability analysis.

Contribution

It introduces a general framework for analyzing eigenvalue accumulation in slow-fast PDEs with canard-like transitions, connecting spectral limits to the absolute spectrum.

Findings

Eigenvalues accumulate onto curves as the slow scale parameter approaches zero.

The limit sets of eigenvalues relate to the absolute spectrum of the rest states.

Results apply to general systems with suitable slow-fast structures.

Abstract

Motivated by pulse-replication phenomena observed in the FitzHugh--Nagumo equation, we investigate traveling pulses whose slow-fast profiles exhibit canard-like transitions. We show that the spectra of the PDE linearization about such pulses may contain many point eigenvalues that accumulate onto a union of curves as the slow scale parameter approaches zero. The limit sets are related to the absolute spectrum of the homogeneous rest states involved in the canard-like transitions. Our results are formulated for general systems that admit an appropriate slow-fast structure.