Nonspiral excitation waves beyond the eikonal approximation

Exact nonplanar traveling solutions for the kinematic model of autowaves in excitable media are constructed using a nonlinear velocity-curvature relation that is more realistic than the linear (eikonal) approximation. The physically unrealistic self-crossing fronts, found earlier in the eikonal approximation, are replaced by noncrossing fronts with cusplike singularities that are reminiscent of ''cellular'' flame structures. The new solutions put restrictions on possible autowave refraction and reflection regimes found in the eikonal approximation. A possible role of the unstable branch of the velocity-curvature relation in removing the cusplike singularities from the present solutions is discussed.