A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation

José Luis López

doi:10.3934/dcds.2020376

Article Contents

2021, Volume 41, Issue 6: 2601-2617. Doi: 10.3934/dcds.2020376

This issue Previous Article Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems Next Article Second order estimates for complex Hessian equations on Hermitian manifolds

A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation

José Luis López

Departamento de Matemática Aplicada and Excellence Research Unit "Modeling Nature" (MNat), Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Received: March 2020

Revised: September 2020

Early access: November 2020

Published: June 2021

The author is partially supported by the MINECO-Feder (Spain) research grant number RTI2018-098850-B-I00, as well as by the Junta de Andalucía (Spain) Project PY18-RT-2422 & A-FQM-311-UGR18.

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Abstract

The parabolic-parabolic Keller-Segel model of chemotaxis is shown to come up as the hydrodynamic system describing the evolution of the modulus square $ n(t,x) $ and the argument $ S(t,x) $ of a wavefunction $ \psi = \sqrt{n} \, e^{iS} $ that solves a cubic Schrödinger equation with focusing interaction, frictional Kostin nonlinearity and Doebner-Goldin dissipation mechanism. This connection is then exploited to construct a family of quasi-stationary solutions to the Keller-Segel system under the influence of no-flux and anti-Fick laws.

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Mathematics Subject Classification: Primary: 35-XX; Secondary: 35Q40, 35K57, 35J60.

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