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

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.

Exact Traveling Wave Solutions of One-Dimensional Parabolic–Parabolic Models of Chemotaxis

“…This system where u = u(y), v = v(y) and λ is an integration constant. As shown in [29] this system passes the Painlevé ODE test only if α = 2 and β = 0. Consequently, in this case we can solve (28 * ) and the exact solution has the form [29]:…”

Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis

“…where u ¼ uy ðÞ , v ¼ vy ðÞ , and λ is an integration constant. As shown in [35], this system passes the Painlevé ODE test only if α ¼ 2 and β ¼ 0. Let us focus on this case.…”

Conservation laws, symmetry reductions, and exact solutions of some Keller–Segel models

“…Keller-Segel models are used to describe a wide range of processes in biology, ecology, medicine, and so on. The readers are referred to [3][4][5][6][7][8][9][10][11][12][13][14] for more details about biological motivation and mathematical introduction of Eq. (1.1).…”

“…System (1.1) with linear law φ(v) = v, ε = 1, f (u, v) = βuλv has the form ⎧ ⎨ ⎩ u t = du xxδ(uv x ) x , v t = αv xx + βuλv, (1.2) and is also called the minimum chemotaxis model, see a review article [6]. When d = δ = 1, λ = 0, system (1.2) becomes of the form in [8,9]:…”

“…For the integrable case α = 2, an exact solution of (1.3) has been derived in [8], more exact solutions of (1.3) have been derived in [9]. (1.4) and has prominent specific applications.…”