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

Shigeki Matsutani

Abstract

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic $\mathrm{al}_{ab}$ functions of a hyperelliptic curve $X$ genus $g$ as the $\mathrm{al}_{ab}$ functions together with the $\mathrm{al}_a$ functions are a generalization of the Jacobi sn, cn, dn functions. We then demonstrate that the $\mathrm{al}_{ab}$ function is a potential hyperelliptic solution to the nonlinear Schrödinger and complex modified Korteweg-de Vries equations as a natural extension of the hyperelliptic solutions of the modified Korteweg-de Vries equation in terms of the $\mathrm{al}_a$ function.

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

Abstract

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic functions of a hyperelliptic curve genus as the functions together with the functions are a generalization of the Jacobi sn, cn, dn functions. We then demonstrate that the function is a potential hyperelliptic solution to the nonlinear Schrödinger and complex modified Korteweg-de Vries equations as a natural extension of the hyperelliptic solutions of the modified Korteweg-de Vries equation in terms of the function.
Paper Structure (17 sections, 24 theorems, 92 equations, 2 figures)

This paper contains 17 sections, 24 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hyperelliptic curve $X$ and sigma functions
  3. Geometrical setting of hyperelliptic curves
  4. Sigma function and its derivatives
  5. The addition formulae
  6. The ${\mathrm {al}}_a$ and ${\mathrm {al}}_{ab}$ functions and their basic properties
  7. The addition theorems for the ${\mathrm {al}}_a$ and ${\mathrm {al}}_{ab}$ functions
  8. The definitions and basic properties of the ${\mathrm {al}}_a$ and ${\mathrm {al}}_{ab}$ functions
  9. The algebraic and geometric foundations of ${\mathrm {al}}_a$ functions, and their differential identities
  10. Double covering of $X$
  11. Differential identities of the ${\mathrm {al}}_a$ functions
  12. The algebraic and geometric foundations of ${\mathrm {al}}_{ab}$ functions, and their differential identities
  13. Algebraic and geometric properties of ${\mathrm {al}}_{ab}$ functions
  14. Differential relations of ${\mathrm {al}}_{ab}$ functions I
  15. Differential relations of ${\mathrm {al}}_{12}$ functions II
  16. ...and 2 more sections

Key Result

Lemma 2.1

Figures (2)

  • Figure 1: The basis of ${\mathbf H}_1(X,{\mathbb Z})$
  • Figure 2: The double covering $\varpi_{{\widehat{X}},1}: {\widehat{X}}_1 \to X$, $\varpi_X: (w_1, z_1) \mapsto (w_1^2+b_1, z_1w_1)=(x,y)$.

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 34 more