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.
Similar content being viewed by others
References
Belokolos E.D., Bobenko A.I., Enolskii V.Z., Its A.R., Matveev V.B. (1994). Algebro-geometric approach to nonlinear integrable equations. Springer, Berlin, Heidelberg, New York
Bobenko A., Bordag L. (1989). Periodic multiphase solutions to the Kadomtsev-Petviashvili equation. J. Phys. A: Math. Gen. 22:1259
Deconinck B., van Hoeij M. (2001). Computing Riemann matrices of algebraic curves. Physica D 152–153:28
Deconinck, B., Heil, M., Bobenko, A., van Hoeij, M., Schmies, M.: Computing Riemann theta functions. Math. Comput. (in press)
Dubrovin B.A., Flickinger R., Segur H. (1997). Three-phase solutions to the Kadomtsev-Petviashvili equation. Stud. Appl. Math. 99(2):137
Ernst F.J. (1968). New formulation of the axially symmetric gravitational field problem. Phys. Rev. 167:1175
Fay J.D. (1973). Theta-functions on Riemann surfaces. Lecture Notes in Mathematics, Vol.352, Springer, Berlin, Heidelberg
Fornberg B. (1996). A practical guide to pseudospectral methods. Cambridge University Press, Cambridge
Frauendiener J., Klein C. (2004). Hyperelliptic theta-functions and spectral methods. J. Comp. Appl. Math. 167:193
Gianni P., Seppälä M., Silhol R., Trager B. (1998). Riemann surfaces, plane algebraic curves and their period matrices. J. Symb. Comp. 26:789
Hoeij M. (1994). An algorithm for computing an integral basis in an algebraic function field. J. Symb. Comput. 18:353
Klein C., Richter O. (1999). Exact relativistic gravitational field of a stationary counter-rotating dust disks. Phys. Rev. Lett. 83:2884
Korotkin D. (1989). Finite-gap solutions of the stationary axisymmetric Einstein equation. Theor. Math. Phys. 77:1018–1031
Mumford D. (1984). Tata Lectures on Theta II. Birkhäuser, Boston
Novikov S., Manakov S., Pitaevskii L., Zakharov V. (1984). Theory of solitons – the inverse scattering method. Consultants Bureau, New York
Seppälä M. (1994). Computation of period matrices of real algebraic curves. Discrete Comput. Geom. 11:65
Tretkoff C.L., Tretkoff M.D. (1984). Combinatorial group theory, Riemann surfaces and differential equations. Contemp. Math. 33:467
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Frauendiener, J., Klein, C. Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP Solutions. Lett Math Phys 76, 249–267 (2006). https://doi.org/10.1007/s11005-006-0068-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-006-0068-4