Letters in Mathematical Physics

, Volume 76, Issue 2–3, pp 249–267 | Cite as

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

Article

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-functions spectral methods 

Mathematics Subject Classifications (2000)

14H55 65N35 14H70 14H42 

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References

  1. 1.
    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 YorkMATHGoogle Scholar
  2. 2.
    Bobenko A., Bordag L. (1989). Periodic multiphase solutions to the Kadomtsev-Petviashvili equation. J. Phys. A: Math. Gen. 22:1259CrossRefADSMathSciNetMATHGoogle Scholar
  3. 3.
    Deconinck B., van Hoeij M. (2001). Computing Riemann matrices of algebraic curves. Physica D 152–153:28CrossRefGoogle Scholar
  4. 4.
    Deconinck, B., Heil, M., Bobenko, A., van Hoeij, M., Schmies, M.: Computing Riemann theta functions. Math. Comput. (in press)Google Scholar
  5. 5.
    Dubrovin B.A., Flickinger R., Segur H. (1997). Three-phase solutions to the Kadomtsev-Petviashvili equation. Stud. Appl. Math. 99(2):137CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Ernst F.J. (1968). New formulation of the axially symmetric gravitational field problem. Phys. Rev. 167:1175CrossRefADSGoogle Scholar
  7. 7.
    Fay J.D. (1973). Theta-functions on Riemann surfaces. Lecture Notes in Mathematics, Vol.352, Springer, Berlin, HeidelbergMATHGoogle Scholar
  8. 8.
    Fornberg B. (1996). A practical guide to pseudospectral methods. Cambridge University Press, CambridgeMATHGoogle Scholar
  9. 9.
    Frauendiener J., Klein C. (2004). Hyperelliptic theta-functions and spectral methods. J. Comp. Appl. Math. 167:193CrossRefADSMathSciNetMATHGoogle Scholar
  10. 10.
    Gianni P., Seppälä M., Silhol R., Trager B. (1998). Riemann surfaces, plane algebraic curves and their period matrices. J. Symb. Comp. 26:789CrossRefMATHGoogle Scholar
  11. 11.
    Hoeij M. (1994). An algorithm for computing an integral basis in an algebraic function field. J. Symb. Comput. 18:353CrossRefMATHGoogle Scholar
  12. 12.
    Klein C., Richter O. (1999). Exact relativistic gravitational field of a stationary counter-rotating dust disks. Phys. Rev. Lett. 83:2884CrossRefADSGoogle Scholar
  13. 13.
    Korotkin D. (1989). Finite-gap solutions of the stationary axisymmetric Einstein equation. Theor. Math. Phys. 77:1018–1031CrossRefMathSciNetGoogle Scholar
  14. 14.
    Mumford D. (1984). Tata Lectures on Theta II. Birkhäuser, BostonMATHGoogle Scholar
  15. 15.
    Novikov S., Manakov S., Pitaevskii L., Zakharov V. (1984). Theory of solitons – the inverse scattering method. Consultants Bureau, New YorkGoogle Scholar
  16. 16.
    Seppälä M. (1994). Computation of period matrices of real algebraic curves. Discrete Comput. Geom. 11:65CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Tretkoff C.L., Tretkoff M.D. (1984). Combinatorial group theory, Riemann surfaces and differential equations. Contemp. Math. 33:467MathSciNetGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Institut für Astronomie und AstrophysikUniversität TübingenTübingenGermany
  2. 2.Max-Planck-Institut für PhysikMünchenGermany
  3. 3.Max-Planck-Institut for Mathematics in the SciencesLeipzigGermany

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