Smoke-Ring Solutions of Gierer–Meinhardt System in $\mathbb{R}^{3}$

Abstract

We consider the steady states of the Gierer–Meinhardt system on all of $\mathbb{R}^{3}$: $\varepsilon^{2}\Delta a-a+\frac{a^{p}}{h^{q}}=0$, $\Delta h-h+\frac{a^{m}}{h^{s}}=0$ with an additional restriction $q=p-1$. In the limit $\varepsilon\rightarrow0$, we use formal asymptotics to construct a solution whose activator component a concentrates on a circle. Under the additional constraints $p>1$, $m>0$, and $1<m-s<3$, we find that such a solution exists and is unique. The radius of the circle of concentration is given explicitly in terms of certain integrals. Full numerical computations are shown to support the analytical results.