Semiclassical approximation for the twodimensional Fisher–Kolmogorov–Petrovskii– Piskunov equation with nonlocal nonlinearity in polar coordinates |
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Authors: | A Yu Trifonov A V Shapovalov |
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Institution: | 1.National Research Tomsk Polytechnic University,Tomsk,Russia;2.Tomsk State University,Tomsk,Russia |
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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. |
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Keywords: | |
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