The one-dimensional Fisher-Kolmogorov equation with a nonlocal nonlinearity in a semiclassical approximation

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Abstract

A model of the evolution of a bacterium population based on the Fisher-Kolmogorov equation is considered. For a one-dimensional equation of the Fisher-Kolmogorov type that contains quadratically nonlinear nonlocal kinetics and weak diffusion terms, a general scheme of semiclassically concentrated asymptotic solutions is developed based on the complex WKB-Maslov method. The solution of the Cauchy problem is constructed in the class of semiclassically concentrated functions. In constructing the solutions, an essential part is played by the dynamic set of Einstein-Ehrenfest equations (a set of equations in average and centered moments) derived in this work. The symmetry operators of the equation, the nonlinear evolution operator, and the class of particular asymptotic semiclassical solutions are found.

Original languageEnglish
Pages (from-to)899-911
Number of pages13
JournalRussian Physics Journal
Volume52
Issue number9
DOIs
Publication statusPublished - 1 Dec 2009

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nonlinearity
approximation
operators
Cauchy problem
Einstein equations
bacteria
moments
kinetics
symmetry

Keywords

  • Complex flow
  • Fisher-Kolmogorov equation
  • Semiclassical approximation

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

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abstract = "A model of the evolution of a bacterium population based on the Fisher-Kolmogorov equation is considered. For a one-dimensional equation of the Fisher-Kolmogorov type that contains quadratically nonlinear nonlocal kinetics and weak diffusion terms, a general scheme of semiclassically concentrated asymptotic solutions is developed based on the complex WKB-Maslov method. The solution of the Cauchy problem is constructed in the class of semiclassically concentrated functions. In constructing the solutions, an essential part is played by the dynamic set of Einstein-Ehrenfest equations (a set of equations in average and centered moments) derived in this work. The symmetry operators of the equation, the nonlinear evolution operator, and the class of particular asymptotic semiclassical solutions are found.",
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AB - A model of the evolution of a bacterium population based on the Fisher-Kolmogorov equation is considered. For a one-dimensional equation of the Fisher-Kolmogorov type that contains quadratically nonlinear nonlocal kinetics and weak diffusion terms, a general scheme of semiclassically concentrated asymptotic solutions is developed based on the complex WKB-Maslov method. The solution of the Cauchy problem is constructed in the class of semiclassically concentrated functions. In constructing the solutions, an essential part is played by the dynamic set of Einstein-Ehrenfest equations (a set of equations in average and centered moments) derived in this work. The symmetry operators of the equation, the nonlinear evolution operator, and the class of particular asymptotic semiclassical solutions are found.

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