Isoperimetric deformations of curves on the Minkowski plane

TL;DR

This paper explores how curves on the Minkowski plane can be deformed isoperimetrically, governed by the defocusing mKdV equation, and provides explicit solutions including singular and regular cases with visual illustrations.

Contribution

It formulates the isoperimetric deformation of Minkowski plane curves via the defocusing mKdV equation and constructs explicit solutions using tau functions, including singular and regular cases.

Findings

Explicit formulas for curve motions on Minkowski plane

Presentation of solutions with singular points

Visual illustrations of curve dynamics

Abstract

We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $\tau$ functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class give regular solutions to the defocusing mKdV equation. Some pictures illustrating typical dynamics of the curves are presented.