Multi-Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain

TL;DR

This paper rigorously analyzes the existence and stability of multi-spike solutions in a fractional Gierer-Meinhardt system, revealing how fractional diffusion influences stability thresholds and solution symmetry.

Contribution

It provides the first rigorous proofs of symmetric and asymmetric two-spike solutions and their stability in the fractional Gierer-Meinhardt model, complemented by numerical analysis.

Findings

Fractional diffusion destabilizes or stabilizes single spikes depending on the component.

Asymmetric two-spike solutions are always linearly unstable.

Fractional inhibitor diffusivity expands the parameter range for stable symmetric two-spike solutions.

Abstract

In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions. In particular we rigorously prove the existence of symmetric and asymmetric two-spike solutions using a Lyapunov-Schmidt reduction. The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain $2\times 2$ matrix. Our rigorous results are complemented by formal calculations of $N$-spike solutions using the method of matched asymptotic expansions. In addition, we explicitly consider examples of one- and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds. By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities. Furthermore, when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity. However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.