Exact solutions for the kinematic model of autowaves in two-dimensional excitable media

The kinematic model for autowaves propagating in two-dimensional homogeneous excitable media is shown to be integrable exactly for steady-state patterns of zero topological charge. A whole set of solutions is found and the stability of the solutions is studied. V-shaped waves observed recently in experiments with Belousov-Zhabotinsky reaction correspond to one of the solutions found. Other solutions turn out to be self-crossing: revealing one or an infinite number of loops. It is shown that regular parts of self-crossing solutions describe different parts of steady-state autowave patterns in piecewise excitable media. Recent experiments with autowaves in excitable media with a stripe of enhanced excitability are analyzed and a new stationary wave pattern is predicted to exist in the case of a stripe of reduced excitability.