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

2025/04/25 by Shigeki Matsutani, Matsutani, Shigeki · 1 voice · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AG #nlin.SI

paper · pdf · doi:10.48550/arxiv.2504.18285

arxiv published 2025/04/25 · arxiv updated 2026/01/23

Abstract

The real part of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field 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, \itJ. Geom. Phys \bf215 (2025) 105540). These are a generalization of Euler's elasticae.

Citations

• Euler's original derivation of elastica equation
• Closed real plane curves of hyperelliptic solutions of focusing gauged modified KdV equation of genus three
• Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic σ functions
• Statistical mechanics of elastica for the shape of supercoiled DNA: hyperelliptic elastica of genus three
• On real hyperelliptic solutions of focusing modified KdV equation
• Commuting Hamiltonian flows of curves in real space forms
• The al function of a cyclic trigonal curve of genus three
• Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti
• Hyperelliptic Solutions of Modified Korteweg-de Vries Equation of Genus g: Essentials of Miura Transformation
• Hyperelliptic Loop Solitons with Genus g: Investigations of a Quantized Elastica
• Statistical Mechanics of Elastica on Plane as a Model of Supercoiled DNA-Origin of the MKdV hierarchy-

Cited by

• Geometric, algebraic and analytic properties of hyperelliptic al_ab function of genus g

Discussions

• Shigeki Matsutani: Real hyperelliptic solutions of gauged modified KdV equation of genus $g$ https://arxiv.org/abs/2504.18285 https://arxiv.org/pdf/2504.18285 https://arxiv.org/html/2504.18285 [bsky, 0 points, 1 comments]

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