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

Shigeki Matsutani

TL;DR

This work addresses constructing explicit real plane curves whose tangential angle satisfies the focusing gauged MKdV equation by leveraging genus-3 hyperelliptic curves. It introduces a refined real parameterization of hyperelliptic solutions, derives a coupled real system for the real and imaginary parts of the complex solution, and proves global solvability under specific reality conditions. The authors present Theorem 4.2 for a nice real parameterization and demonstrate, via numerical experiments, closed elastica-like curves beyond Euler's figure-eight. This generalizes Euler's elastica to higher-genus settings and suggests potential links to physical shapes such as DNA configurations, offering a framework for exploring real solutions of the focusing MKdV equation at genus $g>1$.

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.

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

TL;DR

This work addresses constructing explicit real plane curves whose tangential angle satisfies the focusing gauged MKdV equation by leveraging genus-3 hyperelliptic curves. It introduces a refined real parameterization of hyperelliptic solutions, derives a coupled real system for the real and imaginary parts of the complex solution, and proves global solvability under specific reality conditions. The authors present Theorem 4.2 for a nice real parameterization and demonstrate, via numerical experiments, closed elastica-like curves beyond Euler's figure-eight. This generalizes Euler's elastica to higher-genus settings and suggests potential links to physical shapes such as DNA configurations, offering a framework for exploring real solutions of the focusing MKdV equation at genus .

Abstract

The real and imaginary parts of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field 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.

Paper Structure

This paper contains 6 sections, 15 theorems, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Hyperelliptic solutions of the focusing MKdV equation$/{\mathbb C}$ of genus three
  3. Hyperelliptic curves of genus three in angle expression
  4. Real hyperelliptic solutions of the gauged MKdV equation over ${\mathbb R}$
  5. Numerical computations for closed curves
  6. Discussion and Conclusion

Key Result

Proposition 2.1

Mat02b For $((x_1,y_1),\cdots, (x_3,y_3)) \in S^3 X$, the fixed branch point $(b_0, 0)$, and $u:= v( (x_1,y_1),$$\cdots,(x_3,y_3))$, satisfies the MKdV equation over ${\mathbb C}$, where $\partial_{u_i}:= \partial/\partial u_i$ as an differential identity in $S^3 X$ and ${\mathbb C}^3$.

Figures (5)

  • Figure 1: Euler's figure-eight
  • 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: The orbits of each $\varphi_i$ in the quadrature: $k_1> k_2 > k_3>1.0$.
  • Figure 4: The solutions of (\ref{['eq:Xipnt']}) for $t=0$: $(k_1, k_2, k_3) =(2.12, 1.80, 1.0029)$, and the initial condition is $(\varphi_1, \varphi_2, \varphi_3) = (\varphi_{11}^\circ, \varphi_{21}^\circ, \varphi_{21}^\circ)$. (a): $C_Z$, (b): $\psi_{\mathrm r}$ and $\psi_{\mathrm i}^\circ$ and (c): $\partial_s\psi_{\mathrm r}$ and $\partial_{\mathfrak s}\psi_{\mathrm i}=\partial_{u_3{\mathrm r}} \psi_{\mathrm i}$.
  • Figure 5: The solutions of (\ref{['eq:Xipnt']}) for $t=0$: $(k_1, k_2, k_3) =(2.30, 1.85, 1.005)$, and the initial condition is $(\varphi_1, \varphi_2, \varphi_3) = (\varphi_{12}^\circ, \varphi_{22}^\circ, \varphi_{22}^\circ)$. (a): $Z(s)$, (b): $\psi_{\mathrm r}$ and $\psi_{\mathrm i}^\circ$ and (c): $\partial_s\psi_{\mathrm r}$ and $\partial_{\mathfrak s}\psi_{\mathrm i}=\partial_{u_3{\mathrm r}} \psi_{\mathrm i}$.

Theorems & Definitions (25)

  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 15 more