Multi-spike solutions to the one-dimensional subcritical fractional Schnakenberg system

Abstract

Singularly perturbed reaction–diffusion systems such as the Gierer–Meinhardt, Schankenberg, and Gray–Scott models are known to exhibit localized multi-spike solutions in which one of the species concentrates on a discrete collection of well-separated points. A recent focus has been the extension of the analysis, both rigorous and formal, of such localized solutions to the case where classical diffusion is replaced by anomalous diffusion exhibiting Lévy flights. Mathematically, such systems replace the classical Laplacian with the fractional Laplacian (−Δ)s where s is the fractional order. In this paper we consider the formal analysis of multi-spike solutions to the fractional Schnakenberg system with periodic boundary conditions when the fractional order of the activator satisfies 1/4<s1<1 and that of the substrate satisfies 1/2<s2<1. Using the method of matched asymptotic expansions we derive a nonlinear algebraic system that determines the structure of equilibrium solutions, a nonlocal eigenvalue problem that determines its linear stability over a fast, O(1), timescale, as well as a system of ordinary differential equations that determine the dynamics of multi-spike solutions over a slow, O(ɛ−2), timescale. We explicitly construct symmetric multi-spike solutions and derive stability criteria for instabilities arising over both fast and slow timescales. One of the key findings in this paper is that the slow timescale instability thresholds are, barring symmetry considerations, less degenerate than in the classical s2=1 case. In addition we numerically calculate bifurcation diagrams that show how asymmetric multi-spike solutions branch out from symmetric ones at distinct points for solutions with five or more spikes.