Closed Loop Solitons and Sigma Functions: Classical and Quantized Elasticas with Genera One and Two

Relations in a Loop Soliton as a Quantized Elastica

In the previous article (J. Geom. Phys. {\bf 43} (2002) 146), we show the hyperelliptic solutions of a loop soliton as a study of a quantized elastica. This article gives some functional relations in a loop soliton as a quantized elastica.

Hyperelliptic Loop Solitons with Genus g: Investigations of a Quantized Elastica

In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop solitons in a plane, \it i.e., loops whose curvatures obey the modified Korteweg-de Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous article to those of hyperelliptic curves with general genera. It was proved that the tangential angle of loop soliton is expressed by the Weierstrass hyperelliptic al function for a given hyperelliptic curve $y^2 = f(x)$ with genus $g$.

On the Moduli of a quantized loop in P and KdV flows: Study of hyperelliptic curves as an extension of Euler's perspective of elastica I

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli $\CMeP$ of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere $\PP$. Then it is proved that its moduli space is decomposed to a set of equivalent classes determined by flows obeying the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy has a natural topology, it induces topology in the moduli space …

From Euler's elastica to the mKdV hierarchy, through the Faber polynomials

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.

亏格为三的聚焦规范修正KdV方程超椭圆解的闭合实平面曲线

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

本研究将欧拉弹性线推广，针对亏格为三的超椭圆曲线，通过调节模参数与初始条件，构造出满足聚焦规范修正KdV方程、超越欧拉8字形弹性线的闭合实平面曲线。

具有一族平面曲率线的等温环面与面积约束双曲弹性曲线

Isothermic tori with one family of planar curvature lines and area constrained hyperbolic elastica

本文分类了具有一族平面曲率线的等温环面，给出显式theta函数公式，并揭示其与面积约束双曲弹性曲线的联系，为构造紧致Bonnet对提供基础。