Elastic null curve flows, nonlinear $C$-integrable systems, and geometric realization of Cole-Hopf transformations

TL;DR

This paper introduces a new moving frame for null curves in Minkowski space, revealing integrable systems and geometric flows related to Cole-Hopf transformations, expanding understanding of elastic null curve dynamics.

Contribution

It develops a novel gauge-invariant moving frame for null curves, linking geometric flows to integrable systems and generalizing Cole-Hopf transformations in Minkowski space.

Findings

Derived a hereditary recursion operator encoding integrable hierarchies.

Established connections between geometric null curve flows and Burgers, NLS, and wave equations.

Introduced new hierarchies of integrable systems from symmetry analysis.

Abstract

Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a Frenet null frame by a gauge transformation that belongs to the little group contained in the Lorentz group $SO(2,1)$ and provides an analog of the Hasimoto transformation (relating a parallel frame to a Frenet frame for curves in Euclidean space). The Cartan structure equations of the transformed frame are shown to encode a hereditary recursion operator giving a two-component generalization of the recursion operator of Burgers equation, as well as a generalization of the Cole-Hopf transformation. Three different hierarchies of integrable systems are obtained from the various symmetries of this recursion operator. The first hierarchy contains two-component Burgers-type and nonlinear Airy-type systems; the second hierarchy contains novel quasilinear Schr\"odinger-type (NLS) systems; and the third hierarchy contains semilinear wave equations (in two-component system form). Each of these integrable systems is shown to correspond to a geometrical flow of a family of elastic null curves in three-dimensional Minkowski space.