Explicit Flock Solutions for Quasi-Morse potentials

TL;DR

This paper rigorously analyzes flock solutions in particle systems with Quasi-Morse potentials, establishing existence and uniqueness in three dimensions and existence in two, enhancing understanding of pattern formation in collective behavior models.

Contribution

It provides the first rigorous proof of existence and uniqueness of flock profiles for Quasi-Morse potentials in three dimensions and extends understanding through numerical investigation of related potentials.

Findings

Existence and uniqueness of flock profiles in 3D for Quasi-Morse potentials.

Existence of flock profiles in 2D for Quasi-Morse potentials.

Numerical analysis of Morse-like interactions to explore pattern formation.

Abstract

We consider interacting particle systems and their mean-field limits, which are frequently used to model collective aggregation and are known to demonstrate a rich variety of pattern formations. The interaction is based on a pairwise potential combining short-range repulsion and long-range attraction. We study particular solutions, that are referred to as flocks in the second-order models, for the specific choice of the Quasi-Morse interaction potential. Our main result is a rigorous analysis of continuous, compactly supported flock profiles for the biologically relevant parameter regime. Existence and uniqueness is proven for three space dimension, whilst existence is shown for the two-dimensional case. Furthermore, we numerically investigate additional Morse-like interactions to complete the understanding of this class of potentials.