### 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 language | English |
---|---|

Pages (from-to) | 1243-1253 |

Number of pages | 11 |

Journal | Russian Physics Journal |

Volume | 53 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 May 2011 |

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### 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

**Semiclassical approximation for the twodimensional Fisher-Kolmogorov-Petrovskii- Piskunov equation with nonlocal nonlinearity in polar coordinates.** / Trifonov, A. Yu; Shapovalov, Aleksandr Vasilievich.

Research output: Contribution to journal › Article

}

TY - JOUR

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

UR - http://www.scopus.com/inward/record.url?scp=79957957852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957957852&partnerID=8YFLogxK

U2 - 10.1007/s11182-011-9556-9

DO - 10.1007/s11182-011-9556-9

M3 - Article

VL - 53

SP - 1243

EP - 1253

JO - Russian Physics Journal

JF - Russian Physics Journal

SN - 1064-8887

IS - 12

ER -