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


SIGMA 2 (2006), 061, 15 pages      math-ph/0606042      http://dx.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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References

  1. Dodd R.K., Eilbek J.C., Gibbon J.D., Morris H.C., Solitons and nonlinear wave equations, London, Academic Press, 1984.
  2. Glansdorf P., Prigogine I., Thermodynamics of structure, stability and fluctuations, New York, Wiley, 1971.
  3. Ovsiannikov L.V., Group analysis of differential equations, New York, Academic Press, 1982.
  4. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1993.
  5. Andronov A., Leontovich E., Gordon I., Maier A., Qualitative theory of second order dynamic systems, Jerusalem, Israel Program for Scientific Translations, 1971.
  6. Guckenheimer J., Holmes P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, New York, Springer, 1987.
  7. 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.
  8. Nikitin A., Barannyk T., Solitary waves and other solutions for nonlinear heat equations, Cent. Eur. J. Math., 2004, V.2, 840-858, math-ph/0303004.
  9. 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.
  10. Vladimirov V.A., Kutafina E.V., Exact travelling wave solutions of some nonlinear evolutionary equations, Rep. Math. Phys., 2004, V.54, 261-271.
  11. Makarenko A.S., Mathematical modelling of memory effects influence on fast hydrodynamic and heat conduction processes, Control and Cybernetics, 1996, V.25, 621-630.
  12. 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, J. Theor. Biol., 2000, V. 207, 305-316.
  13. 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, V.56, 421-436.
  14. Hassard B.F., Kazarinoff N.D., Wan Y.-H., Theory and applications of the Hopf bifurcation, New York, Springer, 1981.
  15. Korn G.A., Korn T.M., Mathematical handbook, New York, McGraw-Hill, 1968.
  16. Cornille H., Gervois A., Bi-soliton solutions of a weakly nonlinear evolutionary PDEs with quadratic nonlinearity, Phys. D, 1982, V.6, 1-28.
  17. Leontiev A.F., Entire functions and exponential series, Moscow, Nauka, 1983 (in Russian).
  18. Shil'nikov L.P., On one case of the existence of a countable set of periodic movements, Sov. Math. Dokl., 1965, V.6, 163-166.
  19. Shkadov V.Ya., Solitary waves in a layer of viscous liquid, Fluid Dynam., 1977, V.12, 52-55.

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