Nozomu Matsuura

Discrete mKdV and discrete sine-Gordon flows on discrete space curves

In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric condition. The curve is reconstructed from the deformation data by using the Sym–Tafel formula. The isoperimetric equidistant deformation of the space curves does not preserve the torsion in general. However, it is possible to construct the torsion-preserving deformation by tuning the deformation parameters. Further, it is also possible to make an arbitrary choice of the deformation described by the discrete mKdV equation or by the discrete sine-Gordon equation at each step. We finally show that the discrete deformation of discrete space curves yields the discrete K-surfaces.

Discrete Models of Isoperimetric Deformation of Plane Curves

We consider the isoperimetric deformation of smooth curves on the Euclidean plane. It naturally gives rise to a nonlinear partial differential equation called the modified KdV(mKdV) equation as a deformation equation of the curvature, which is known as one of the most typical example of the soliton equations or the integrable systems. The Frenet equation and the deformation equation of the Frenet frame of the curve are the auxiliary linear problem or the Lax pair of the mKdV equation. Based on this formulation, we present two discrete models of isoperimetric deformation of plane curves preserving underlying integrable structure: the discrete deformation described by the discrete mKdV equation and the continuous deformation described by the semi-discrete mKdV equation.

Discrete Isoperimetric Deformation of Discrete Curves

We consider isoperimetric deformations of discrete plane/space curves. We first give a brief review of the theory of isoperimetric deformation of smooth curves, which naturally gives rise to the modified KdV (mKdV) equation as a deformation equation of the curvature. We then present its discrete model described by the discrete mKdV equation, which is formulated as the isoperimetric equidistant deformation of discrete curves. We next give a review of isoperimetric and torsion-preserving deformation of smooth space curves with constant torsion which is described by the mKdV equation. We formulate a discrete analogue of it as the isoperimetric, torsion-preserving and equidistant deformation on the osculating planes of space discrete curves. The deformation admits two discrete flows, namely by the discrete mKdV equation and by the discrete sine-Gordon equation. We also show that one can make an arbitrary choice of two flows at each step, which is controlled by tuning the deformation parameters appropriately.