Explicit formulas for isoperimetric deformations of smooth and discrete elasticae

Abstract

We construct the explicit formula for the isoperimetric deformation of elastica described by the modified KdV equation. We also construct the explicit formulas for the continuous and discrete deformations of the discrete analogue of elastica described by the semi-discrete potential modified KdV equation and the discrete potential modified KdV equation, respectively. The formulas are given in terms of the elliptic theta functions.