Pattern formation in terms of semiclassically limited distribution on lower-dimensional manifolds for nonlocal Fisher--Kolmogorov--Petrovskii--Piskunov equation

TL;DR

This paper studies pattern formation in nonlocal Fisher-KPP equations, deriving integro-differential equations for pattern dynamics on lower-dimensional manifolds and providing asymptotic solutions validated by numerical simulations.

Contribution

It introduces a novel approach using semiclassically limited distributions to analyze pattern formation on lower-dimensional manifolds in nonlocal PDEs.

Findings

Derived integro-differential equations for pattern dynamics.

Obtained asymptotic large-time solutions for steady-state patterns.

Validated analytical results with numerical simulations.

Abstract

We have investigated the pattern formation in systems described by the nonlocal Fisher--Kolmogorov--Petrovskii--Piskunov equation for the cases where the dimension of the pattern concentration area is less than that of independent variables space. We have obtained a system of integro-differential equations which describe the dynamics of the concentration area and the semiclassically limited distribution of a pattern in the class of trajectory concentrated functions. Also, asymptotic large-time solutions have been obtained that describe the semiclassically limited distribution of a quasi-steady-state pattern on the concentration manifold. The approach is illustrated by an example for which the analytical solution is in good agreement with the prediction of a numerical simulation.