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The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: The particular integrable case and soliton solution
P.-H. Chavanis, “Chaos, complexity and transport,” in Proceedings of Fifteenth Annual Symposium on the Theory of Computing, Marseille, 2007.
N. A. Kudryashov, The Analytical Theory of Nonlinear Differential Equations (MEPhI, Moscow, 2002), and references therein.
A. F. Nikiforov and V. B. Uvarov, The Special Functions of Mathematical Physics (Nauka, Moscow1978).
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In this paper, we investigate the one-dimensional parabolic-parabolic Patlak-Keller-Segel model of chemotaxis. For the case when the diffusion coefficient of chemical substance is equal to two, in terms of travelling wave variables the reduced system appears integrable and allows the analytical solution. We obtain the exact soliton solutions, one of which is exactly the one-soliton solution of the Korteweg-de Vries equation.
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