Differential Equations of Genus Four Hyperelliptic $\wp$ Functions

Polynomial Dynamical Systems and Differentiation of Genus 4 Hyperelliptic Functions

We give an explicit solution to the problem of differentiation of hyperelliptic functions in genus 4 case. We describe explicitly the polynomial Lie algebras and polynomial dynamical systems connected to this problem.

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。

$\mathbb{C}^\infty$中多项式向量场确定任意亏格双曲函数的微分

Polynomial vector fields in $\mathbb{C}^\infty$ determining differentiation of hyperelliptic functions of any genus

本文通过直接证明，展示了$\mathbb{C}^\infty$中多项式向量场与任意亏格双曲函数微分之间的关系，证明了相关算子的交换性及多项式湮灭性质。

Differentiation of Genus $4$ Hyperelliptic Functions

In this work we give an explicit solution to the problem of differentiation of hyperelliptic functions in genus $4$ case. It is a genus $4$ analogue of the classical result of F. G. Frobenius and L. Stickelberger [F. G. Frobenius, L. Stickelberger, "Uber die Differentiation der elliptischen Functionen nach den Perioden und Invarianten", J. Reine Angew. Math., 92 (1882), 311-337] in the case of elliptic functions. An explicit solution in the genus $2$ case was given in [V. M. Buchstaber, "Polynomial dynamical systems and Korteweg-de Vries equation", Proc. Steklov Inst. Math., 294 (2016), 176-20…

广义mKdV方程与二 genus 面积型双曲方程

Generalized mKdV Equation and Genus Two Jacobi Type Hyperelliptic Differential Equation

本文提出广义mKdV方程，研究了二 genus 双曲曲线下的Weierstrass型和Jacobi型双曲函数，并通过Sp(4,R)变换连接其解。

Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3

Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyp…

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus $3$ is described in terms of the gradient of it's sigma function. Solutions of corresponding families of polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals are constructed as an application. These systems were introduced in the work of V. M. Buchstaber and A. V. Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves.

超螺旋DNA形状的弹性线统计力学：亏格三的超椭圆弹性线

Statistical mechanics of elastica for the shape of supercoiled DNA: hyperelliptic elastica of genus three

本文用弹性线统计力学模型研究超螺旋DNA形状，通过聚焦修正KdV方程描述热涨落导致的S-8字形激发态，成功再现了AFM观测到的超螺旋DNA形态。

Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Equations of dispersionless Hirota type have been thoroughly investigated in the mathematical physics and differential geometry literature. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional and the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the structure of the `master-equation' corresponding to this orbit remained elusive. Here we prove that the master-equation is specified by the vanishing of any genus 3 theta constant with even characteristic. The rich geometry of integrable Hirota type eq…

双曲曲线al_ab函数的几何、代数和分析性质

Geometric, algebraic and analytic properties of $\mathrm{al}_{ab}$ function for hyperelliptic curves of genus $g$

本文研究双曲曲线X的al_ab函数的几何、代数和分析性质，这些函数是Jacobi椭圆函数的推广，并推导出新的可积非线性偏微分方程。

Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian

The sine-Gordon equation has hyperelliptic al function solutions over a hyperelliptic Jacobian for $y^2 = f(x)$ of arbitrary genus $g$. This article gives an extension of the sine-Gordon equation to that over subvarieties of the hyperelliptic Jacobian. We also obtain the condition that the sine-Gordon equation in a proper subvariety of the Jacobian is defined.

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigma-function of genus two. For each sequence there is an associated genus two curve $X$, such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of $X$. The construction is based on a Lax pair with a spectral curve $S$ of genus four admitting an involution $σ$ with two fixed points, and the Jacobian of $X$ arises as the Prym variety Prym$(S,σ)$.

Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP solutions

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristic quantities of the Riemann surface with high precision even in almost degenerate situations as in the solitonic limit where the branch points coincide pairwise. As an example we consider hyperelliptic s…