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

TL;DR

This paper analyzes a specific integrable case of the 1D chemotaxis model, deriving exact soliton solutions including a KdV soliton, advancing understanding of chemotactic pattern formation.

Contribution

It identifies an integrable case of the 1D Patlak-Keller-Segel model and derives explicit soliton solutions, including a KdV soliton, which was not previously known.

Findings

Exact soliton solutions for the integrable case obtained

One solution corresponds to the Korteweg-de Vries soliton

The model's reduced system is integrable under specific conditions

Abstract

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.