Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 061, 15 pages      math-ph/0606042      https://doi.org/10.3842/SIGMA.2006.061

Constructing Soliton and Kink Solutions of PDE Models in Transport and Biology

Vsevolod A. Vladimirov, Ekaterina V. Kutafina and Anna Pudelko
Faculty of Applied Mathematics AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland

Received November 30, 2005, in final form May 24, 2006; Published online June 19, 2006

Abstract
We present a review of our recent works directed towards discovery of a periodic, kink-like and soliton-like travelling wave solutions within the models of transport phenomena and the mathematical biology. Analytical description of these wave patterns is carried out by means of our modification of the direct algebraic balance method. In the case when the analytical description fails, we propose to approximate invariant travelling wave solutions by means of an infinite series of exponential functions. The effectiveness of the method of approximation is demonstrated on a hyperbolic modification of Burgers equation.

Key words: generalized Burgers equation; telegraph equation; model of somitogenesis; direct algebraic balance method; periodic and solution-like travelling wave solutions; approximation of the soliton-like solutions.

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