### Abstract

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

Original language | English |
---|---|

Article number | 1850102 |

Journal | International Journal of Geometric Methods in Modern Physics |

Volume | 15 |

Issue number | 6 |

DOIs | |

Publication status | Accepted/In press - 28 Feb 2018 |

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### Keywords

- complex germ
- Nonlocal Fisher–KPP equation
- pattern formation
- semiclassical approximation
- symmetry operators

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

**An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation.** / Shapovalov, A. V.; Trifonov, A. Yu.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

AU - Shapovalov, A. V.

AU - Trifonov, A. Yu

PY - 2018/2/28

Y1 - 2018/2/28

N2 - A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

AB - A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

KW - complex germ

KW - Nonlocal Fisher–KPP equation

KW - pattern formation

KW - semiclassical approximation

KW - symmetry operators

UR - http://www.scopus.com/inward/record.url?scp=85042745012&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042745012&partnerID=8YFLogxK

U2 - 10.1142/S0219887818501025

DO - 10.1142/S0219887818501025

M3 - Article

AN - SCOPUS:85042745012

VL - 15

JO - International Journal of Geometric Methods in Modern Physics

JF - International Journal of Geometric Methods in Modern Physics

SN - 0219-8878

IS - 6

M1 - 1850102

ER -