Single-spike solutions to the 1D shadow Gierer-Meinhardt problem

TL;DR

This paper derives exact one-dimensional spike solutions for the shadow Gierer-Meinhardt reaction-diffusion system using a novel hyperbolic function ansatz, confirming and extending previous numerical findings.

Contribution

It introduces a new analytical method to find exact solutions for the shadow Gierer-Meinhardt problem, overcoming limitations of standard asymptotic techniques.

Findings

Derived exact radially symmetric solutions for 1D shadow Gierer-Meinhardt system.

Confirmed numerical results from previous studies.

Provided a framework for extending solutions to different boundary conditions and higher dimensions.

Abstract

A fundamental example of reaction-diffusion system exhibiting Turing type pattern formation is the Gierer-Meinhardt system, which reduces to the shadow Gierer-Meinhardt problem in a suitable singular limit. Thanks to its applicability in a large range of biological applications, this singularly perturbed problem has been widely studied in the last few decades via rigorous, asymptotic, and numerical methods. However, standard matched asymptotics methods do not apply (Ni 1998, Wei 1998), and therefore analytical expressions for single spike solutions are generally lacking. By introducing an ansatz based on generalized hyperbolic functions, we determine exact radially symmetric solutions to the one-dimensional shadow Gierer-Meinhardt problem for any $1 < p < \infty$, representing both inner and boundary spike solutions depending on the location of the peak. Our approach not only confirms numerical results existing in literature, but also provides guidance for tackling extensions of the shadow Gierer-Meinhardt problem based on different boundary conditions (e.g. mixed) and/or $n$-dimensional domains.