On the structure of steady solutions for the kinematic model of spiral waves in excitable media
The subject of this paper is a rigorous analysis of stationary solutions to the kinematic equation which phenomenologically describes the propagation of a reaction front in an excitable medium. The phenomenological propagation model postulates the local normal velocity of the front propagation to be given by the expression \(V = V_0 -Dk\), where \(V_0\) and \(D\) are constants and \(k\) is the local curvature of the fronts. This description is a sort of ``nonlinear Huygens' principle'' for propagating waves in excitable media (``autowaves''). For the case of stationary propagation, the description can be finally transformed into a form of an equation of motion for a Newtonian particle in a potential well, oscillating in the presence of a friction force with an effective friction coefficient that is a function of the particle's coordinate: \[ d^{2}l/ds^2 = (V_0/D)e^l - e^{2l} + (\omega/D)(dl/ds)e^{-l}, \] where \(l\equiv \ln k(s)\), \(s\) is the length along the front, and \(\omega\) is an arbitrary constant (generated by the integration of the underlying kinematic equation). Analyzing the latter equation, the authors classify its different solutions according to the value of \(\omega\) and initial conditions, which, eventually, allows them to interpret the solutions geometrically as various asymptotic forms of the kinematically evolving reaction front. In particular, the Archimedean spiral can be obtained only in special cases, and other cases give rise to non-Archimedean spirals.
scientific article; zbMATH DE number 1187606