Beyond all order asymptotics for homoclinic snaking in a Schnakenberg system

Abstract

Abstract Using multi-scale beyond all order methods, we investigate stationary spatially localized solutions of a Schnakenberg system, a prototype reaction–diffusion system, near the onset of a subcritical Turing bifurcation. These solutions are homoclinic orbits to a homogeneous solution and passing near a periodic solution. In bifurcation diagrams, branches of these solutions commonly show two interwining snaking curves. Here we calculate the maximal range of existence for these solutions and compare our findings with numerical computations. We derive and optimally truncate a (divergent) asymptotic series of a front solution. The remainder of the truncated series is exponentially small if and only if a specific parameter range is met. This complements work on Swift–Hohenberg equations where similar results have been obtained.