SIGMA 2 (2006), 061, 15 pages math-ph/0606042
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
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.
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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- Dodd R.K., Eilbek J.C., Gibbon J.D., Morris H.C., Solitons and
nonlinear wave equations, London, Academic Press, 1984.
- Glansdorf P., Prigogine I., Thermodynamics of structure, stability
and fluctuations, New York, Wiley, 1971.
- Ovsiannikov L.V., Group analysis of differential equations, New
York, Academic Press, 1982.
- Olver P., Applications of Lie groups to differential equations,
New York, Springer, 1993.
- Andronov A., Leontovich E., Gordon I., Maier A., Qualitative
theory of second order dynamic systems, Jerusalem, Israel Program
for Scientific Translations, 1971.
- Guckenheimer J., Holmes P., Nonlinear oscillations, dynamical
systems and bifurcations of vector fields, New York, Springer,
- Fan E., Multiple travelling wave solutions of nonlinear evolution
equations using a unified algebraic method, J. Phys. A:
Math. Gen., 2002, V.35, 6853-6872.
- Nikitin A., Barannyk T., Solitary waves and other solutions for
nonlinear heat equations, Cent. Eur. J. Math., 2004, V.2,
- Barannyk A., Yurik I., Construction of exact solutions of
diffusion equations, in Proceedings of Fifth International
Conference "Symmetry in Nonlinear Mathematical Physics" (June
23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O.
Popovych and I.A. Yehorchenko, Proceedings of Institute of
Mathematics, Kyiv, 2004, V.50, Part 1, 29-33.
- Vladimirov V.A., Kutafina E.V., Exact travelling wave solutions
of some nonlinear evolutionary equations, Rep. Math. Phys.,
2004, V.54, 261-271.
- Makarenko A.S., Mathematical modelling of memory effects
influence on fast hydrodynamic and heat conduction processes,
Control and Cybernetics, 1996, V.25, 621-630.
- Collier J.D., McInerney D., Schnell S., Maini P.K., Gavaghan
D.J., Houston P., Stern C.D., A cell cyclic model for
somitogenesis: mathematical formulation and numerical simulation,
Biol., 2000, V. 207, 305-316.
- Vladimirov V.A., Kutafina E.V., Toward an approximation of
solitary-wave solutions of non-integrable evolutionary PDEs via
symmetry and qualitative analysis, Rep. Math. Phys., 2005,
- Hassard B.F., Kazarinoff N.D., Wan Y.-H., Theory and applications of
the Hopf bifurcation, New York, Springer, 1981.
- Korn G.A., Korn T.M., Mathematical handbook, New York,
- Cornille H., Gervois A., Bi-soliton solutions of a weakly
nonlinear evolutionary PDEs with quadratic nonlinearity,
Phys. D, 1982, V.6, 1-28.
- Leontiev A.F., Entire functions and exponential series, Moscow, Nauka, 1983 (in Russian).
- Shil'nikov L.P., On one case of the existence of a countable set
of periodic movements, Sov. Math. Dokl., 1965, V.6,
- Shkadov V.Ya., Solitary waves in a layer of viscous liquid, Fluid Dynam., 1977, V.12, 52-55.