Computation of rotating wave solutions of reaction-diffusion equations on a circle (Q2785700)

A control theoretical approach to compute nonuniform periodic solutions which rotate with a nonzero angular velocity is described. The corresponding reaction-diffusion equation has the form \(\frac{\partial u_j}{\partial t}= D_j \frac{\partial^2u_j}{\partial \theta^2}+f_j (u_1, u_2,\dots,u_m;\mu)\) on \(S\times(0,\infty)\), where \(D_1,D_2,\dots,D_m\) are positive constants, and \(\theta\) is the angular variable on the unit circle \([0,2\pi]\). The solutions are of the form \(u_j(\theta,t)=v_j(\theta-ct)\).NEWLINENEWLINENEWLINENumerical methods for computing these waves are described. Their bifurcation behavior is mentioned. A least squares method using a control theoretic version of shooting to find the solution is shown. Four steps of the conjugate gradient algorithm are given. The numerical computation of rotating waves on a circle is illustrated on some solutions for the ``Brusselator''.