Differential Geometry of the Vortex Filament Equation

TL;DR

This paper develops differential calculus on asymptotically linear curves and applies it to analyze the vortex filament equation, revealing its Hamiltonian structure, recursion operator, and constants of motion.

Contribution

It introduces a differential calculus framework for asymptotically linear curves and demonstrates its application to the vortex filament equation's Hamiltonian properties.

Findings

Recursion operator is hereditary.

The system has a Hamiltonian pair structure.

Constants of motion are explicitly derived.

Abstract

Differential calculus on the space of asymptotically linear curves is developed. The calculus is applied to the vortex filament equation in its Hamiltonian description. The recursion operator generating the infinite sequence of commuting flows is shown to be hereditary. The system is shown to have a description with a Hamiltonian pair. Master symmetries are found and are applied to deriving an expression of the constants of motion in involution. The expression agrees with the inspection of Langer and Perline.