Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

TL;DR

This paper derives explicit solutions for the motion of discrete plane curves using semi-discrete and discrete potential modified KdV equations, connecting discrete and smooth curve motions through tau functions and Bäcklund transformations.

Contribution

It provides explicit formulas for discrete curve motion solutions and explores their continuous limits, advancing understanding of discrete integrable geometry.

Findings

Explicit solutions in terms of tau functions

Bäcklund transformations for discrete curves

Continuous limit connecting discrete and smooth curves

Abstract

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $\tau$ function are presented. B\"acklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.