Letters in Mathematical Physics

, Volume 76, Issue 2, pp 249-267

First online:

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

  • J. FrauendienerAffiliated withInstitut für Astronomie und Astrophysik, Universität Tübingen
  • , C. KleinAffiliated withMax-Planck-Institut für PhysikMax-Planck-Institut for Mathematics in the Sciences Email author 

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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.


hyperelliptic theta-functions spectral methods

Mathematics Subject Classifications (2000)

14H55 65N35 14H70 14H42