Letters in Mathematical Physics

, Volume 76, Issue 2, pp 249–267

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

Authors

  • J. Frauendiener
    • Institut für Astronomie und AstrophysikUniversität Tübingen
    • Max-Planck-Institut für Physik
    • Max-Planck-Institut for Mathematics in the Sciences
Article

DOI: 10.1007/s11005-006-0068-4

Cite this article as:
Frauendiener, J. & Klein, C. Lett Math Phys (2006) 76: 249. doi:10.1007/s11005-006-0068-4

Abstract

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 solutions to the Kadomtsev–Petviashvili and the Korteweg–de Vries equations. Tests of the numerics using identities for periods on the Riemann surface and the differential equations are performed. It is shown that an accuracy of the order of machine precision can be achieved.

Keywords

hyperelliptic theta-functionsspectral methods

Mathematics Subject Classifications (2000)

14H5565N3514H7014H42

Copyright information

© Springer 2006