# Hyperelliptic theta-functions and spectral methods

@article{Frauendiener2004HyperellipticTA, title={Hyperelliptic theta-functions and spectral methods}, author={J{\"o}rg Frauendiener and Christian Klein}, journal={Journal of Computational and Applied Mathematics}, year={2004}, volume={167}, pages={193-218} }

## Figures from this paper

## 32 Citations

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

- Mathematics
- 2006

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 Theta-functions and Spectral Methods Ii

- Mathematics
- 2004

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…

Computational Approach to Hyperelliptic Riemann Surfaces

- Mathematics
- 2014

We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is given by a list of the branch points, the coefficients of the defining…

COMPUTATIONAL APPROACH TO HYPERELLIPTIC

- Mathematics
- 2014

We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is either given by a list of the branch points, the coefficients of the defining…

Computational approach to compact Riemann surfaces

- Mathematics
- 2015

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton…

Algebraic Curves and Riemann Surfaces in Matlab

- Mathematics
- 2011

In the previous chapter, a detailed description of the algorithms for the ‘algcurves’ package in Maple was presented. As discussed there, the package is able to handle general algebraic curves with…

New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2012

An independent derivation of solutions to the Camassa–Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay’s identities. Reality and smoothness…

Computation of the topological type of a real Riemann surface

- MathematicsMath. Comput.
- 2014

An algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve is presented, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution, and whether this set divides the surface into one or two connected components.

Efficient Computation of the Branching Structure of an Algebraic Curve

- MathematicsArXiv
- 2011

An efficient algorithm for computing the branching structure of a compact Riemann surface defined via an algebraic curve is presented. Generators of the fundamental group of the base of the ramified…

Computational approach to Riemann surfaces

- Mathematics
- 2011

Introduction to Compact Riemann Surfaces.- Computing with plane algebraic curves and Riemann surfaces: the algorithms of the Maple package "algcurves".- Algebraic curves and Riemann surfaces in…

## References

SHOWING 1-10 OF 43 REFERENCES

Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces

- Mathematics
- 1998

We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Meinel and Neugebauer can be…

Riemann Surfaces, Plane Algebraic Curves and Their Period Matrices

- Mathematics, Computer ScienceJ. Symb. Comput.
- 1998

The design of an algorithm which computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve thus solves an important problem in the general case.

Exact relativistic treatment of stationary counterrotating dust disks: Boundary value problems and solutions

- Mathematics, Physics
- 2001

This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counterrotating disks of dust. These disks can serve…

Computing Riemann theta functions

- Mathematics, Computer ScienceMath. Comput.
- 2004

Algorithms for its computation are given and a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision.

NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES

- Mathematics
- 1976

The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical…

Algebro-geometric approach to nonlinear integrable equations

- Mathematics, Physics
- 1994

A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving…

On the exact relativistic treatment of stationary counter-rotating dust disks IIIPhysical Properties

- Physics
- 2001

This is the third in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We…

Theta functions and non-linear equations

- Mathematics
- 1981

CONTENTS Introduction Chapter I. Theta functions. General information § 1. Definition of theta functions and their simplest properties § 2. Theta functions of a single variable § 3. On Abelian tori §…

Exact Solutions of Einstein's Field Equations

- Physics
- 2004

We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in Negi and Durgapal [Gravitation and Cosmology7, 37 (2001)] which should be…