Integrable dynamics of a discrete curve and the Ablowitz–Ladik hierarchy

Abstract

It is shown that the following elementary geometric properties of the motion of a discrete (i.e., piecewise linear) curve select the integrable dynamics of the Ablowitz–Ladik hierarchy of evolution equations: (i) the set of points describing the discrete curve lie in the sphere S3; (ii) the distance between any two subsequent points does not vary in time; (iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. These results generalize to a discrete context the previous work on continuous curves [A. Doliwa and P. M. Santini, Phys. Lett. A 185, 373 (1994)].