Integrable systems from inelastic curve flows in 2- and 3- dimensional Minkowski space

TL;DR

This paper derives integrable systems from inelastic curve flows in Minkowski space using a Lorentzian moving frame, revealing connections to well-known equations like mKdV, NLS, and Burgers' equation with new geometric insights.

Contribution

It extends the geometrical moving frame method to Minkowski space, deriving new integrable equations and their bi-Hamiltonian structures from inelastic curve flows.

Findings

Timelike/spacelike flows yield defocusing mKdV and NLS equations.

Null flows produce Burgers' equation and symmetry structures.

Spacelike flows in 3D Minkowski space involve hyperbolic numbers.

Abstract

Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2- and 3- dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear Schrodinger (NLS) equation in 2- and 3- dimensional Euclidean space, respectively. In 2-dimensional Minkowski space, timelike/spacelike inelastic curve flows are shown to yield the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while inelastic null curve flows are shown to give rise to Burgers' equation and its symmetry integrability structure. In 3-dimensional Minkowski space, the complex defocusing mKdV equation and the NLS equation along with their bi-Hamiltonian integrability structures are obtained from timelike inelastic curve flows, whereas spacelike inelastic curve flows yield an interesting variant of these two integrable equations in which complex numbers are replaced by hyperbolic (split-complex) numbers.