On traveling waves of generalized chemotaxis models of Keller-Segel type inspired in Doebner-Goldin theory

Abstract

The standard parabolic-parabolic Keller-Segel model of chemotaxis, along with some of its most well-known variants, e.g. those including a logistic growth term or a logarithmic sensitivity function in the cell equation, are shown to come up as the hydrodynamic systems describing the evolution of the modulus square n(t, x) and the argument S(t, x) of a complex wavefunction ψ = √ neiS that solves a cubic Schrödinger equation with focusing interaction, frictional Kostin nonlinearity and Doebner-Goldin dissipation mechanism. This connection is then exploited to construct some important families of traveling-wave solutions.