Infinitesimal Darboux transformation and semi-discrete mKdV equation

TL;DR

This paper explores the connection between continuous motions of discrete planar curves and the semi-discrete potential mKdV equation through Darboux transformations, introducing infinitesimal transformations and a geometric interpretation.

Contribution

It introduces infinitesimal Darboux transformations and provides a geometric perspective linking discrete curve motions to the semi-discrete mKdV equation.

Findings

Establishes a link between continuous curve motions and semi-discrete integrable equations.

Defines infinitesimal Darboux transformations including curve motions.

Provides a geometric interpretation for the semi-discrete potential mKdV equation.

Abstract

We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg-de Vries (mKdV) equation with Darboux transformations of smooth planar curves. In doing so, we define infinitesimal Darboux transformations that include the aforementioned motions, and also give an alternate geometric interpretation for establishing the semi-discrete potential mKdV equation.