On the Existence of Leapfrogging Pair of Circular Vortex Filaments

TL;DR

This paper develops a mathematical model for the motion of vortex filaments, deriving a precise condition under which leapfrogging occurs, and proves its necessity and sufficiency.

Contribution

It introduces a new system of nonlinear PDEs for vortex filament motion and establishes a rigorous condition for leapfrogging in coaxial circular filaments.

Findings

Derived a condition for leapfrogging to occur.

Proved the condition is necessary and sufficient.

Provided a mathematical framework for vortex filament dynamics.

Abstract

We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. Furthermore, for a pair of coaxial circular vortex filaments, we derive a condition for leapfrogging to occur and prove that the condition is necessary and sufficient for the occurrence of leapfrogging.