Spiral Waves in Reaction-Diffusion Equations

We consider the reaction-diffusion system \[\begin{gathered} R_T = \nabla ^2 R + R\left( {1 - R^2 - \vec \nabla \theta \cdot \vec \nabla \theta } \right), \hfill \\ R\theta _T = R\nabla ^2 \theta + 2\vec \nabla R \cdot \vec \nabla \theta + qR^3 \hfill \\ \end{gathered} \]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 \cdots $. We also find the explicit one-dimensional analogues of spiral waves for all q, and show that they are unstable for $| q |$ small.