An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher-KPP equation. (English) Zbl 1458.35432
Summary: 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.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B36 | Pattern formations in context of PDEs |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 76M60 | Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 92D25 | Population dynamics (general) |
| 35K58 | Semilinear parabolic equations |
Keywords:
nonlocal Fisher-KPP equation; semiclassical approximation; complex germ; symmetry operators; pattern formationReferences:
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