Skip to main content
SpringerLink
Log in

Computational Approach to Hyperelliptic Riemann Surfaces

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is 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 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Belokolos E.D., Bobenko A.I., Enolskii V.Z., Its A.R., Matveev V.B.: Algebro-geometric approach to nonlinear integrable equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  2. Bianchi, L.: Lezioni di Geometria Differenziale, (Pisa 1909)

  3. Bobenko, A.I., Klein, C. (eds.): Computational approach to riemann surfaces, Lect. Notes Math. 2013 (2011)

  4. Bour, E.: Théorie de la déformation des surfaces. J. Ecole Imperiale Polytechnique 19, 1:48 (1862)

  5. Deconinck B., van Hoeij M.: Computing Riemann matrices of algebraic curves. Phys. D 152153, 28 (2001)

    Article  Google Scholar 

  6. Deconinck B., Heil M., Bobenko A., van Hoeij M., Schmies M.: Computing Riemann theta functions. Math. Comput. 73, 1417–1442 (2004)

    Article  ADS  MATH  Google Scholar 

  7. Deconinck, B., Patterson, M.: Computational Approach to Riemann Surfaces. In: Bobenko, A.I., Klein, C. (eds.), Lect. Notes Math. 2013 (2011)

  8. Dubard P., Gaillard P., Klein C., Matveev V.B.: On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Special Topics 185, 247258 (2010)

    Article  Google Scholar 

  9. Dubrovin, B.A.: Theta functions and non-linear equations, Usp. Mat. Nauk 36(2), 11–80 (1981) (English translation: Russ. Math. Surv. 36(2), 11–92 (1981))

  10. Enolski V.Z., Richter P.H.: Periods of hyperelliptic integrals expressed in terms of θ-constants by means of Thomae formulae. Phil. Trans. R. Soc. A 366, 1005–1024 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Faddeev L.D., Korepin V.E.: Quantum theory of solitons. Phys. Rep. 42(1), 187 (1978)

    Article  MathSciNet  Google Scholar 

  12. Faltings G.: Calculus on arithmetic surfaces. Ann. Math. 119, 387–424 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Frauendiener J., Klein C.: On the exact treatment of stationary counter-rotating dust disks: physical properties. Phys. Rev. D 63, 84025 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  14. Frauendiener J., Klein C.: Hyperelliptic theta-functions and spectral methods. J. Comput. Appl. Math. 167, 193 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Frauendiener J., Klein C.: Hyperelliptic theta-functions and spectral methods: KdV and KP solutions. Lett. Math. Phys. 76, 249–267 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Frauendiener, J., Klein, C.: Computational Approach to Riemann Surfaces. In: Bobenko, A.I., Klein, C. (eds.) Lect. Notes Math. 2013 (2011)

  17. Grava T., Klein C.: Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations. Commun. Pure Appl. Math. 60, 1623–1664 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Grava T., Tian F.-R.: The generation, propagation, and extinction of multiphases in the KdV zero-dispersion limit. Commun. Pure Appl. Math. 55(12), 1569–1639 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. de Jong, R.: Admissible constants for genus two curves. Bull. LMS 42, 405–411 (2010)

  20. Kalla, C., Klein, C.: New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation. Proc. R. Soc. A. doi:10.1098/rspa.2011.0583. (2012)

  21. Kamvissis, S., McLaughlin, K.D.T.-R. Miller, P.D., Semiclassical soliton ensembles for the focusing nonlinear schrödinger equation. Ann. Math. Stud. vol. 154. Princeton University Press, Princeton (2003)

  22. Klein C., Kokotov A., Korotkin D.: Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces. Math. Zeitschr. 261(1), 73–108 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Klein, C., Richter, O.: Ernst equation and Riemann surfaces, vol. 685, Lecture Notes in Physics, Springer, Berlin (2005)

  24. Korotkin. D.A.: Finite-gap solutions of the stationary axially symmetric Einstein equation in vacuum. Theor. Math. Phys. 77, 1018 (1989)

  25. Korotkin D.: On some integrable cases in surface theory. J. Math. Sci. 94, 1177 (1999)

    Article  MathSciNet  Google Scholar 

  26. Bobenko A., Bordag L.: Periodic multiphase solutions to the Kadomtsev-Petviashvili equation. J. Phys. A. Math. Gen. 22, 1259 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Kozel, V.A., Kotlyarov, V.P.: Almost periodical solutions to the equation U TT −U XX  + sinU = 0, In: Kotlyarov, V.P. (ed.) Finite-gap solutions of the sine-Gordon equation, Dopovidi Akademii NAUK Ukrainskoi A-Fiz. Mat., vol. 10, pp. 878–881 (1976) (English translation and summary of orginal papers at arXiv:1401.4410)

  28. Kowalewski S.: Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 14, 8193 (1889)

    Google Scholar 

  29. Lax, P.D., Levermore, C.D.: The small dispersion limit of the Korteweg de Vries equation, I, II, III. Commun. Pure Appl. Math. 36, 253–290, 571–593, 809–830 (1983)

  30. Mumford, D.: Tata lectures on theta. II, volume 29 of Progress in Mathematics. Birkhäuser, Boston (1983)

  31. Neumann, C.: J. Reine Angew. Math. 56, 46–53 (1859)

  32. Quine, J.R., Sarnak, P. (eds.): Extremal Riemann surfaces. Contemp. Math., 201 AMS (1997)

  33. Trefethen L.N.: Spectral methods in Matlab. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  34. www.comlab.ox.ac.uk/oucl/work/nick.trefethen

  35. Zeng Z.: Computing multiple roots of inexact polynomials. Math. Comput. 74, 869–903 (2004)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9010, New Zealand

    Jörg Frauendiener

  2. Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078, Dijon Cedex, France

    Christian Klein

Authors
  1. Jörg Frauendiener
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Klein.

Additional information

This work has been supported in part by the Marie-Curie IRSES project RIMMP and the Royal Society of New Zealand.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frauendiener, J., Klein, C. Computational Approach to Hyperelliptic Riemann Surfaces. Lett Math Phys 105, 379–400 (2015). https://doi.org/10.1007/s11005-015-0743-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-015-0743-4

Mathematics Subject Classification

Keywords

Search

Navigation