Geodesic
Constructing Soliton and Kink Solutions of PDE Models in Transport and Biology
Symmetry, integrability and geometry: methods and applications, Tome 2 (2006) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: 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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     title = {Constructing {Soliton} and {Kink} {Solutions} of {PDE} {Models} in {Transport} and {Biology}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a60/}
}
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Vsevolod A. Vladimirov; Ekaterina V. Kutafina; Anna Pudelko. Constructing Soliton and Kink Solutions of PDE Models in Transport and Biology. Symmetry, integrability and geometry: methods and applications, Tome 2 (2006). http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a60/

[1] Dodd R. K., Eilbek J. C., Gibbon J. D., Morris H. C., Solitons and nonlinear wave equations, Academic Press, London, 1984

[2] Glansdorf P., Prigogine I., Thermodynamics of structure, stability and fluctuations, Wiley, New York, 1971 | Zbl

[3] Ovsiannikov L. V., Group analysis of differential equations, Academic Press, New York, 1982 | MR | Zbl

[4] Olver P., Applications of Lie groups to differential equations, Springer, New York, 1993 | MR

[5] Andronov A., Leontovich E., Gordon I., Maier A., Qualitative theory of second order dynamic systems, Israel Program for Scientific Translations, Jerusalem, 1971 | MR

[6] Guckenheimer J., Holmes P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, New York, 1987 | MR

[7] Fan E., “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method”, J. Phys. A: Math. Gen., 35 (2002), 6853–6872 | DOI | MR | Zbl

[8] Nikitin A., Barannyk T., “Solitary waves and other solutions for nonlinear heat equations”, Cent. Eur. J. Math., 2 (2004), 840–858 ; arXiv:math-ph/0303004 | DOI | MR | Zbl

[9] Barannyk A., Yurik I., “Construction of exact solutions of diffusion equations”, Proceedings of Fifth International Conference “Symmetry in Nonlinear Mathematical Physics”, Part 1 (June 23–29, 2003, Kyiv), Proceedings of Institute of Mathematics, 50, eds. A. G. Nikitin, V. M. Boyko, R. O. Popovych and I. A. Yehorchenko, Kyiv, 2004, 29–33 | MR | Zbl

[10] Vladimirov V. A., Kutafina E. V., “Exact travelling wave solutions of some nonlinear evolutionary equations”, Rep. Math. Phys., 54 (2004), 261–271 | DOI | MR | Zbl

[11] Makarenko A. S., “Mathematical modelling of memory effects influence on fast hydrodynamic and heat conduction processes”, Control and Cybernetics, 25 (1996), 621–630 | MR | Zbl

[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., 207 (2000), 305–316 | DOI

[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., 56 (2005), 421–436 | DOI | MR | Zbl

[14] Hassard B. F., Kazarinoff N. D., Wan Y.-H., Theory and applications of the Hopf bifurcation, Springer, New York, 1981 | MR

[15] Korn G. A., Korn T. M., Mathematical handbook, McGraw-Hill, New York, 1968 | MR

[16] Cornille H., Gervois A., “Bi-soliton solutions of a weakly nonlinear evolutionary PDEs with quadratic nonlinearity”, Phys. D, 6 (1982), 1–28 | DOI | MR

[17] Leontiev A. F., Entire functions and exponential series, Nauka, Moscow, 1983 (in Russian)

[18] Shil'nikov L. P., “On one case of the existence of a countable set of periodic movements”, Sov. Math. Dokl., 6 (1965), 163–166

[19] Shkadov V. Ya., “Solitary waves in a layer of viscous liquid”, Fluid Dynam., 12 (1977), 52–55 | DOI | Zbl