Semiclassical approximation for the twodimensional Fisher-Kolmogorov-Petrovskii- Piskunov equation with nonlocal nonlinearity in polar coordinates

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Abstract

The two-dimensional Kolmogorov-Petrovskii-Piskunov-Fisher equation with nonlocal nonlinearity and axially symmetric coefficients in polar coordinates is considered. The method of separation of variables in polar coordinates and the nonlinear superposition principle proposed by the authors are used to construct the asymptotic solution of a Cauchy problem in a special class of smooth functions. The functions of this class arbitrarily depend on the angular variable and are semiclassically concentrated in the radial variable. The angular dependence of the function has been exactly taken into account in the solution. For the radial equation, the formalism of semiclassical asymptotics has been developed for the class of functions which singularly depend on an asymptotic small parameter, whose part is played by the diffusion coefficient. A dynamic system of Einstein-Ehrenfest equations (a system of equations in mean and central moments) has been derived. The evolution operator for the class of functions under consideration has been constructed in explicit form.

Original languageEnglish
Pages (from-to)1243-1253
Number of pages11
JournalRussian Physics Journal
Volume53
Issue number12
DOIs
Publication statusPublished - 1 May 2011

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polar coordinates
nonlinearity
approximation
Cauchy problem
Einstein equations
diffusion coefficient
formalism
moments
operators
coefficients

Keywords

  • evolution operator
  • Fisher-Kolmogorov-Petrovskii-Piskunov equation
  • nonlinear superposition principle
  • nonlocal nonlinearity
  • semiclassical asymptotics

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

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title = "Semiclassical approximation for the twodimensional Fisher-Kolmogorov-Petrovskii- Piskunov equation with nonlocal nonlinearity in polar coordinates",
abstract = "The two-dimensional Kolmogorov-Petrovskii-Piskunov-Fisher equation with nonlocal nonlinearity and axially symmetric coefficients in polar coordinates is considered. The method of separation of variables in polar coordinates and the nonlinear superposition principle proposed by the authors are used to construct the asymptotic solution of a Cauchy problem in a special class of smooth functions. The functions of this class arbitrarily depend on the angular variable and are semiclassically concentrated in the radial variable. The angular dependence of the function has been exactly taken into account in the solution. For the radial equation, the formalism of semiclassical asymptotics has been developed for the class of functions which singularly depend on an asymptotic small parameter, whose part is played by the diffusion coefficient. A dynamic system of Einstein-Ehrenfest equations (a system of equations in mean and central moments) has been derived. The evolution operator for the class of functions under consideration has been constructed in explicit form.",
keywords = "evolution operator, Fisher-Kolmogorov-Petrovskii-Piskunov equation, nonlinear superposition principle, nonlocal nonlinearity, semiclassical asymptotics",
author = "Trifonov, {A. Yu} and Shapovalov, {Aleksandr Vasilievich}",
year = "2011",
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T1 - Semiclassical approximation for the twodimensional Fisher-Kolmogorov-Petrovskii- Piskunov equation with nonlocal nonlinearity in polar coordinates

AU - Trifonov, A. Yu

AU - Shapovalov, Aleksandr Vasilievich

PY - 2011/5/1

Y1 - 2011/5/1

N2 - The two-dimensional Kolmogorov-Petrovskii-Piskunov-Fisher equation with nonlocal nonlinearity and axially symmetric coefficients in polar coordinates is considered. The method of separation of variables in polar coordinates and the nonlinear superposition principle proposed by the authors are used to construct the asymptotic solution of a Cauchy problem in a special class of smooth functions. The functions of this class arbitrarily depend on the angular variable and are semiclassically concentrated in the radial variable. The angular dependence of the function has been exactly taken into account in the solution. For the radial equation, the formalism of semiclassical asymptotics has been developed for the class of functions which singularly depend on an asymptotic small parameter, whose part is played by the diffusion coefficient. A dynamic system of Einstein-Ehrenfest equations (a system of equations in mean and central moments) has been derived. The evolution operator for the class of functions under consideration has been constructed in explicit form.

AB - The two-dimensional Kolmogorov-Petrovskii-Piskunov-Fisher equation with nonlocal nonlinearity and axially symmetric coefficients in polar coordinates is considered. The method of separation of variables in polar coordinates and the nonlinear superposition principle proposed by the authors are used to construct the asymptotic solution of a Cauchy problem in a special class of smooth functions. The functions of this class arbitrarily depend on the angular variable and are semiclassically concentrated in the radial variable. The angular dependence of the function has been exactly taken into account in the solution. For the radial equation, the formalism of semiclassical asymptotics has been developed for the class of functions which singularly depend on an asymptotic small parameter, whose part is played by the diffusion coefficient. A dynamic system of Einstein-Ehrenfest equations (a system of equations in mean and central moments) has been derived. The evolution operator for the class of functions under consideration has been constructed in explicit form.

KW - evolution operator

KW - Fisher-Kolmogorov-Petrovskii-Piskunov equation

KW - nonlinear superposition principle

KW - nonlocal nonlinearity

KW - semiclassical asymptotics

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