# Standing waves of fixed period for $n+1$ vortex filaments

## TL;DR

This paper investigates standing wave patterns in vortex filament configurations, revealing infinite bifurcating solutions at specific period ratios, expanding understanding of vortex dynamics.

## Contribution

It introduces new standing wave solutions bifurcating from known rotating vortex configurations at rational period ratios.

## Key findings

- Existence of infinite standing wave solutions
- Bifurcation from uniform rotating configurations
- Dependence on rational period ratios

## Abstract

The $n+1$ vortex filament problem has explicit solutions consisting of $n$ parallel filaments of equal circulation in the form of nested polygons uniformly rotating around a central filament which has circulation of opposite sign. We show that when the relation between temporal and spatial periods is fixed at certain rational numbers, these configurations have an infinite number of homographic time dependent standing wave patterns that bifurcate from these uniformly rotating central configurations.