The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: the particular integrable case and soliton solution

摘要

In this paper we investigate the one-dimensional parabolic-parabolicPatlak-Keller-Segel model of chemotaxis. For the case when the diffusioncoefficient of chemical substance is equal to two, in terms of travelling wavevariables the reduced system appears integrable and allows the analyticalsolution. We obtain the exact soliton solutions, one of which is exactly theone-soliton solution of the Korteweg-de Vries equation.