An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation

TL;DR

This paper develops a semiclassical approximation method using the Maslov complex germ approach for the 1D nonlocal Fisher-KPP equation, enabling explicit construction of solutions and analysis of large-time behavior.

Contribution

It introduces a novel semiclassical formalism for the Fisher-KPP equation, reducing it to linear equations and algebraic conditions, and constructs solutions valid over finite and large times.

Findings

Explicit semiclassical asymptotics for the nonlinear equation

Solutions exhibit transition from unimodal to multimodal distributions

Method provides a way to analyze large-time evolution of solutions

Abstract

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov 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-Kolmogorov-Petrovskii-Piskunov 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.