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.
In: Russian Physics Journal, Vol. 53, No. 12, 01.05.2011, p. 1243-1253.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 -