Self-similar Solutions for the Kinematic Model Equation of Spiral Waves

We present a class of self-similar solutions of the kinematic model equation, introduced by V.A. Davydov, A.S. Mikhailov, and V.S. Zykov. This equation is designed to describe the dynamics of spiral waves in excitable media. In this model the sharply located spiral fronts are regarded as planar curves. If the tip neither grows nor retracts in the tangential direction and if their normal velocity (with the eikonal approximation) is assumed to possess no driving force, then the kinematic equation admits self-similar solutions with nonzero curvature. We show the global structure of both forward and backward self-similar solutions, which implies mathematically the existence of various types of spiral waves.