MATHEMATICAL BIOSCIENCES http://www.mbejournal.org/ AND ENGINEERING Volume 5, Number 1, January 2008 pp. 205–216 SOLITON-LIKE EXCITATION IN A NONLINEAR MODEL OF DNA DYNAMICS WITH VISCOSITY Conrad Bertrand Tabi Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P. O. Box 812, Yaounde, Cameroon Alidou Mohamadou Condensed Matter Physics Laboratory, Department of Physics Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon Timoleon Crepin Kofane Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P. O. Box 812, Yaounde, Cameroon (Communicated by Stefano Boccaletti) Abstract. The study of solitary wave solutions is of prime significance for nonlinear physical systems. The Peyrard-Bishop model for DNA dynamics is generalized specifically to include the difference among bases pairs and vis- cosity. The small amplitude dynamics of the model is studied analytically and reduced to a discrete complex Ginzburg-Landau (DCGL) equation. Ex- act solutions of the obtained wave equation are obtained by the mean of the extended Jacobian elliptic function approach. These amplitude solutions are made of bubble solitons. The propagation of a soliton-like excitation in a DNA is then investigated through numerical integration of the motion equations. We show that discreteness can drastically change the soliton shape. The impact of viscosity as well as elasticity on DNA dynamic is also presented. The pro- file of solitary wave structures as well as the energy which is initially evenly distributed over the lattice are displayed for some fixed parameters. 1. Introduction. The local opening of DNA is an intriguing phenomenon from a statistical-physics point of view, but is also essential for its biological function. For instance, the transcription and replication of our genetic code cannot take place without the unwinding of the DNA double helix. Although these biological pro- cesses are driven by protein, there might well be a relation between these biological opening processes and the spontaneous bubble formation. The key problem in DNA biophysics is how to relate functional properties of DNA with its structural and physical dynamical characteristics. The local openings can be analytically described as breather-like objects of small amplitude, which nevertheless have in- teresting properties: as long as their amplitude is small enough, they can move along the chain, collect energy and grow [1]. They can also be trapped by some 2000 Mathematics Subject Classification. 37N25, 34J60, 34G34 . Key words and phrases. DNA dynamics, discrete complex Ginzburg-Landau equation, Jaco- bian elliptic functions, bubble soliton, density energy. 205