Smoke-Ring Solutions of Gierer-Meinhardt System in R

We consider the steady states of the Gierer-Meinhardt system on all of R-3: epsilon(2)Delta a - a + a(p)/h(q) = 0, Delta h - h + a(m)/h(s) = 0 with an additional restriction q = p - 1. In the limit e. 0, 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.