Commun. Comput. Phys., 8 (2010), pp. 610-622. |
On Spiral Waves Arising in Natural Systems D. Bini ^{1*}, C. Cherubini ^{2}, S. Filippi ^{2}, A. Gizzi ^{3}, P. E. Ricci ^{4} 1 Istituto per le Applicazioni del Calcolo "M. Picone", CNR I-00161 Rome, Italy; and International Center for Relativistic Astrophysics -I.C.R.A., University of Rome "La Sapienza'', I-00185 Rome, Italy.2 International Center for Relativistic Astrophysics -I.C.R.A., University of Rome "La Sapienza'', I-00185 Rome, Italy; and Nonlinear Physics and Mathematical Modeling Lab, Faculty of Engineering, University Campus Bio-Medico, I-00128 Rome, Italy. 3 Nonlinear Physics and Mathematical Modeling Lab, Faculty of Engineering, University Campus Bio-Medico, I-00128 Rome, Italy; and Alberto Sordi Foundation-Research Institute on Aging, I-00128 Rome, Italy. 4 Mathematics Department, University of Rome "La Sapienza'', I-00185 Rome, Italy. Received 24 September 2009; Accepted (in revised version) 17 February 2010 Available online 15 April 2010 doi:10.4208/cicp.240909.170210a Abstract Spiral waves appear in many different natural contexts: excitable biological tissues, fungi and amoebae colonies, chemical reactions, growing crystals, fluids and gas eddies as well as in galaxies. While the existing theories explain the presence of spirals in terms of nonlinear parabolic equations, it is explored here the fact that self-sustained spiral wave regime is already present in the linear heat operator, in terms of integer Bessel functions of complex argument. Such solutions, even if commonly not discussed in the literature because diverging at spatial infinity, play a central role in the understanding of the universality of spiral process. In particular, we have studied how in nonlinear reaction-diffusion models the linear part of the equations determines the wave front appearance while nonlinearities are mandatory to cancel out the blowup of solutions. The spiral wave pattern still requires however at least two cross-reacting species to be physically realized. Biological implications of such a results are discussed. AMS subject classifications: 92B25, 33E99, 65Z05PACS: 05.45.-a, 87.10.-e, 51.20.+d, 98.62.Hr Key words: Reaction-diffusion equations, biophysics, crystal growth, heat transfer. *Corresponding author. Email: binid@icra.it (D. Bini), c.cherubini@unicampus.it (C. Cherubini), s.filippi@unicampus.it (S. Filippi), a.gizzi@unicampus.it (A. Gizzi), paoloemilio.ricci@uniroma1.it (P. E. Ricci) |