Closed real plane curves of hyperelliptic solutions of focusing gauged modified KdV equation of genus three

TL;DR

This paper investigates closed real plane curves related to hyperelliptic solutions of the focusing gauged MKdV equation of genus three, revealing new curve shapes beyond classical elastica through parameter tuning.

Contribution

It introduces a study of hyperelliptic genus three solutions of the FGMKdV equation and demonstrates the existence of novel closed plane curves beyond Euler's elastica.

Findings

Existence of closed real plane curves beyond classical elastica.

Parameter tuning yields diverse curve shapes.

Hyperelliptic solutions of genus three are explicitly linked to curve geometry.

Abstract

The real and imaginary parts of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field $\mathbb{C}$ give rise to the focusing gauged MKdV (FGMKdV) equations. As a generalization of Euler's elastica whose curvature obeys the focusing static MKdV (FSMKdV) equation, we study real plane curves whose curvature obeys the FGMKdV equation since the FSMKdV equation is a special case of the FGMKdV equation. In this paper, we focus on the hyperelliptic curves of genus three. By tuning some moduli parameters and initial conditions, we show closed real plane curves associated with the FGMKdV equation beyond Euler's figure-eight of elastica.