Patterns with prescribed numbers of critical points on topological tori

TL;DR

This paper investigates the existence and construction of stable pattern solutions with specified numbers of critical points on topological tori, providing explicit examples and locations.

Contribution

It introduces a method to construct topological tori and patterns with prescribed critical points, advancing understanding of pattern formation on complex surfaces.

Findings

Explicit construction of patterns with prescribed critical points

Demonstration of stable nonconstant stationary solutions on tori

Locations of critical points are explicitly determined

Abstract

We study the existence of critical points of stable stationary solutions to reaction-diffusion problems on topological tori. Stable nonconstant stationary solutions are often called patterns. We construct topological tori and patterns with prescribed numbers of critical points whose locations are explicit.