# Existence and Stability of Four-Vortex Collinear Relative Equilibria with Three Equal Vorticities

## TL;DR

This paper investigates the existence, classification, and stability of collinear relative equilibria in a four-vortex system with three equal vorticities, using algebraic geometry and bifurcation analysis.

## Contribution

It provides a comprehensive analysis of solutions and stability for four-vortex configurations with three equal vorticities, including bifurcation and stability results.

## Key findings

- Stable solutions exist with mixed sign vorticities.
- Solutions are organized into two groups based on vortex ordering.
- Bifurcation analysis reveals solution structure changes.

## Abstract

We study collinear relative equilibria of the planar four-vortex problem where three of the four vortex strengths are identical. The $S_3$ invariance obtained from the equality of vorticities is used to reduce the defining equations and organize the solutions into two distinct groups based on the ordering of the vortices along the line. The number and type of solutions are given, along with a discussion of the bifurcations that occur. The linear stability of all solutions is investigated rigorously and stable solutions are found to exist for cases where the vorticities have mixed signs. We employ a combination of analysis and computational algebraic geometry to prove our results.