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

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.

💡 Research Summary

The paper investigates elastic (stretching) flows of null curves in three‑dimensional Minkowski space ( \mathbb{R}^{2,1} ). A central contribution is the introduction of a new “null‑tangent” moving frame that does not rely on the pseudo‑arclength parameter. This frame is related to the standard Frenet null frame by a gauge transformation belonging to the little group of the Lorentz group (SO(2,1)). The transformation plays the role of a Minkowski‑space analogue of the Hasimoto map, converting the Frenet curvature‑torsion pair ((\kappa,\tau)) into a pair of complex potentials ((u,v)).

The Cartan matrix of the null‑tangent frame encodes a two‑component hereditary recursion operator (\mathcal{R}). By imposing a natural geometric gauge condition, the authors obtain a two‑component Cole‑Hopf transformation that linearizes the system: the potentials ((u,v)) are expressed as logarithmic derivatives of two independent scalar fields ((\phi,\psi)) satisfying decoupled heat equations (\phi_t=\phi_{xx}), (\psi_t=\psi_{xx}). Consequently, the nonlinear evolution equations for ((u,v)) inherit integrability from the linear heat flow.

Three distinct symmetry families of (\mathcal{R}) generate three hierarchies of integrable two‑component nonlinear PDEs:

1. First hierarchy – contains a two‑component Burgers‑type system
   \