Moving null curves and integrability

Abstract

We study the null curves and their motion in a $3$-dimensional flat space-time $M_{3}$. We show that when the motion of null curves forms two surfaces in $M_{3}$ the integrability conditions lead to the well-known AKNS hierarchy. In this case we obtain all the geometrical quantities of the surfaces arising from the whole hierarchy but we particulary focus on the surfaces of the MKdV and KdV equations. We obtain one- and two-soliton surfaces associated to the MKdV equation and show that the Gauss and mean curvatures of these surfaces develop singularities in finite time. We show that the tetrad vectors on the curves satisfy the spin vector equation in the ferromagnetism model of Heisenberg.