Hamiltonian flows on null curves

TL;DR

This paper explores the dynamics of null curves in Minkowski space, linking their evolution to integrable systems like the KdV hierarchy and Painlevé equations, revealing geometric and physical insights.

Contribution

It establishes a connection between null curve motions and integrable equations, introducing special motions and analyzing shape-preserving flows in Minkowski space.

Findings

Null curves evolving under KdV flows are characterized.

Shape-preserving null curve motions relate to a particle model with linear curvature Lagrangian.

Curvature evolution under similarity transformations involves Painlevé II solutions.

Abstract

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, it is shown that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painlev\'e equation.