In this paper the authors discuss the connection between local motion of a null curve in the three-dimensional Minkowski space \(\mathbb{L}^3\) and the KdV equation, extending in this way previous work by E. Musso and L. Nicolodi [ibid. 23, No. 9, 2117–2129 (2010; Zbl 1203.37116)].
More concretely, they study the evolution equation, named the null localized induction equation, given by \[ \gamma_t=- k T -2 N\, ,\tag{NLIE} \] for null curves \(\gamma (\sigma,t)\) in the \(3\)-dimensional Minkowski space. Here \(\sigma\) represents the pseudo-arc-length parameter, \(\{T,W,N\}\) denotes the Cartan frame for a null curve and \(\kappa\) stands for its curvature.
Inspired by Gelfand and L. A. Dickey’s approach to integrability [Soliton equations and Hamiltonian systems. 2nd ed. Singapore: World Scientific (2003; Zbl 1140.35012)], they give a Lie algebra structure on variation vector fields along a null curve. Then, they establish a one-to-one homomorphism which identifies a certain Lie subalgebra of variation vector fields with a Lie subalgebra of derivation vector fields, which is used to construct an explicit Hamiltonian structure for null curve evolution in \(\mathbb{L}^3\).
Finally, they obtain the NLIE as a Hamiltonian evolution equation for the total curvature Hamiltonian \(\int \kappa\, d\sigma\) and, using the previous mentioned one-to-one homomorphism, they connect the NLIE and KdV phase spaces, from which the integrability of the NLIE is deduced as a direct consequence of the integrability of the KdV equation.
The same type of arguments have been previously used by the authors to study the integrability of the “planar filament equation” [J. Geom. Phys. 88, 94–104 (2015; Zbl 1315.14047)], what suggests that this approach could be successfully extended to a more general procedure conducting to a better understanding of the relations between integrable PDE’s and the geometric motions of curves.