Spiral Waves in Reaction-Diffusion Equations

We consider the reaction-diffusion system \[{gathered} R_T = ∇ ^2 R + R( {1 - R^2 - ∇ θ · ∇ θ } ), \\ Rθ _T = R∇ ^2 θ + 2 ∇ R · ∇ θ + qR^3 \\ {gathered} \]This system governs the solutions of reaction-diffusion systems near a Hopf bifurcation point. In two spatial dimensions we use formal asymptotic expansions to contruct one-armed and multi-armed Archimedean spiral waves for small values of the parameter q. We then show that the one-armed spiral waves are probably stable and the multi-armed ones are unstable for $| q |$ small. Next, by numerical continuation methods, we construct spiral waves for all q. These calculations show that the one-armed spiral waves are unstable when | q | > 1.397 ⋯. We also find the explicit one-dimensional analogues of spiral waves for all q, and show that they are unstable for $| q |$ small.