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

TL;DR

The paper develops a real algebro-geometric framework to construct hyperelliptic solutions of the focusing gauged MKdV equation (FGMKdV) associated with genus $g$ hyperelliptic curves, extending Euler’s elastica to higher-genus closed plane curves. The core method couples a double covering ${\widehat{X}}\to X$ and an angle-expression formalism, employing shifted elementary symmetric polynomials to translate Abel–Jacobi data into real solutions, with explicit genus $g=5$ analysis and full generalization to arbitrary $g$. Local and global reality results are established for genus $g>2$, and numerical experiments for genus $5$ produce several closed elastica-like curves (Models A–D), demonstrating richer geometric shapes and gauge effects. The work links integrable systems, algebraic curves, and differential geometry of elastica to potential biological applications such as DNA supercoiling, while outlining future directions in moduli classification and higher-genus extensions.

Abstract

The real part of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field $\mathbb{C}$ is reduced to the focusing gauged MKdV (FGMKdV) equation. In this paper, we construct the real hyperelliptic solutions of FGMKdV equation in terms of data of the hyperelliptic curves of genus $g$ and demonstrate the closed hyperelliptic plane curves of genus $g=5$ whose curvature obeys the FGMKdV equation by extending the previous results of genus three (Matsutani, {\it{J. Geom. Phys}} {\bf{215}} (2025) 105540). These are a generalization of Euler's elasticae.

Figures (10)

• Figure 1: Euler's figure-eight and previous result in in Ma24d: (a): Euler's figure-eight Euler44, (b) $Z(s)$, (c): $\psi_{\mathrm r}$ and $\psi_{\mathrm i}^\circ$. (d): $\partial_s\psi_{\mathrm r}$ and $\partial_{\mathfrak s}\psi_{\mathrm i}=\partial_{u_3{\mathrm r}} \psi_{\mathrm i}$.
• Figure 2: The double covering $\varpi_{\widehat{X}}: {\widehat{X}} \to X$, $\varpi_X: (w, z) \mapsto (w^2+b_0, zw)=(x,y)$.
• Figure 3: ${\mathcal{A}}_X \subset S^1$: $k_1> k_2 > \cdots > k_g>1.0$ for the odd $g$ (a) and the even $g$ (b).
• Figure 4: ${\mathcal{A}}_X \cup H_{\mathbb R} \subset {\mathbb C} \times {\mathbb R}$ for the odd $g$ ($g=5$) (a), and the even $g$ ($g=4$) (b).
• Figure 5: The solutions of (\ref{['eq:Xipnt']}) for $t=0$ of model A: (a): $C_Z$, (b): $\psi_{\mathrm r}$ and $\psi_{\mathrm i}^\circ$ and (c): $\partial_s\psi_{\mathrm r}$, $\partial_{\mathfrak s}\psi_{\mathrm i}=\partial_{u_3{\mathrm r}} \psi_{\mathrm i}$, and $\varphi=\psi_{\mathrm r}/2-\varphi_0$ with $(\varphi_a)_{a=1,\ldots,5}$.

Theorems & Definitions (58)

• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• Definition 3.5
• Lemma 3.6
• proof