On the structure of steady solutions for the kinematic model of spiral waves in excitable media
type: Item
instance of: scholarly article
title: On the structure of steady solutions for the kinematic model of spiral waves in excitable media
zbMATH Open document ID: 0904.92002
DOI: 10.1007/BF03167407
author: Ryo Ikota
author: Naoyuki Ishimura
author: Tomohiko Yamaguchi
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 1998-10-25T00:00:00Z
review text: 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.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 92B05
Mathematics Subject Classification ID: 76Z99
zbMATH DE Number: 1187606
zbMATH Keywords: autowaves
zbMATH Keywords: Archimedean spirals
zbMATH Keywords: Huygens' principle
zbMATH Keywords: kinematic equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1187606