Computational approach to hyperelliptic Riemann surfaces

TL;DR

This paper develops a computational method for analyzing hyperelliptic Riemann surfaces in Weierstrass form, enabling precise calculations of periods, Abel maps, and handling degenerate cases, with applications to integrable systems.

Contribution

It introduces an algorithmic approach to compute homology bases, periods, and Abel maps for general hyperelliptic surfaces, extending previous methods to more complex cases.

Findings

Spectral accuracy achieved with Clenshaw-Curtis method

Handles nearly degenerate Riemann surfaces

Applied to solutions of the sine-Gordon equation

Abstract

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 polynomial or a system of cuts for the curve. A canonical basis of the homology is introduced algorithmically for this curve. The periods of the holomorphic differentials and the Abel map are computed with the Clenshaw-Curtis method in order to achieve spectral accuracy. The code can handle almost degenerate Riemann surfaces. This work generalizes previous work on real hyperelliptic surfaces with prescribed cuts to arbitrary hyperelliptic surfaces. As an example, solutions to the sine-Gordon equation in terms of multi-dimensional theta functions are studied, also in the solitonic limit of these solutions.