Bäcklund Transformations and New Integrable Systems on the Plane

  • A. V. TsiganovEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 273)


The hyperelliptic curve cryptography is based on the arithmetic in the Jacobian of a curve. In classical mechanics well-known cryptographic algorithms and protocols can be very useful for construct auto-Bäcklund transformations, discretization of continuous flows and study of integrable systems with higher order integrals of motion. We consider application of a standard arithmetic of divisors on genus two hyperelliptic curve for the construction of new auto-Bäcklund transformations for the Hénon-Heiles system. Another type of auto-Bäcklund transformations associated with equivalence relations between unreduced divisors and the construction of the new integrable systems in the framework of the Jacobi method are also discussed.


Bäcklund transformations Integrable systems Hyperelliptic curve cryptography 


  1. 1.
    Abel, N.H.: Mémoire sure une propriété générale d’une class très éntendue des fonctions transcendantes, pp. 145–211. Œuvres complétes, Tom I, Grondahl Son, Christiania (1881)Google Scholar
  2. 2.
    Ballesteros, A., Blasco, A., Herranz, F.J., Musso, F.: An integrable Hénon-Heiles system on the sphere and the hyperbolic plane. Nonlinearity 28, 3789–3801 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bolsinov, A.V., Jovanović, B.: Integrable geodesic flows on Riemannian manifolds: construction and obstructions. In: Contemporary Geometry and Related Topics, pp. 57–103. World Scientific Publishing, River Edge, NJ (2004)Google Scholar
  4. 4.
    Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comput. 48(177), 95–101 (1987)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Costello, C., Lauter, K.: Group law computations on Jacobians of hyperelliptic curves. In: Miri, A., Vaudenay, S. (eds.) SAC 2011. LNCS, vol. 7118, pp. 92–117. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Dubrovin, B.A.: Riemann surfaces and nonlinear equations. Russ. Math. Surv. 36(2), 11–93 (1981)CrossRefGoogle Scholar
  7. 7.
    Fedorov, Y.: Discrete versions of some algebraic integrable systems related to generalized Jacobians. In: SIDE III: Symmetries and Integrability of Difference Equations, (Sabaudia, 1998), CRM Proceedings. Lecture Notes, vol. 25, pp. 147–160. American Mathematical Society, Providence, RI (2000)Google Scholar
  8. 8.
    Galbraith, S.G., Harrison, M., Mireles Morales, D.J.: Efficient hyperelliptic arithmetic using balanced representation for divisors. In: van der Poorten, A.J., Stein, A. (eds.) Algorithmic Number Theory 8th International Symposium (AN TS-VIII). Lecture Notes in Computer Science, vol. 5011, pp. 342–356. Springer, Heidelberg (2008)Google Scholar
  9. 9.
    Gantmacher, F.: Lectures in Analytical Mechanics. Mir Publishers, Moscow (1975)Google Scholar
  10. 10.
    Gaudry P., Harley R.: Counting points on hyperelliptic curves over finite fields. In: Bosma, W. (ed.) ANTS. Lecture Notes in Computer Science, vol. 1838, pp. 313–332. Springer, Heidelberg (2000)Google Scholar
  11. 11.
    Grigoryev, Y.A., Sozonov, A.P., Tsiganov, A.V.: Integrability of Nonholonomic Heisenberg Type Systems, SIGMA, v. 12, vol. 112, p. 14 (2016)Google Scholar
  12. 12.
    Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman and Hall/CRC, Boca Raton (2006)Google Scholar
  13. 13.
    Harley R.: Fast arithmetic on genus two curves (2000).
  14. 14.
    Hietarinta, J.: Direct methods for the search of the second invariant. Phys. Rep. 147, 87–154 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hone, A.N.W., Kuznetsov, V.B., Ragnisco, O.: Bäcklund transformations for many-body systems related to KdV. J. Phys. A Math. Gen. 32, L299–L306 (1999)CrossRefGoogle Scholar
  16. 16.
    Inoue, R., Konishi, Y., Yamazaki, T.: Jacobian variety and integrable system—after Mumford, Beauville and Vanhaecke. J. Geom. Phys. 57(3), 815–831 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jacobi, C.G.J.: Über eine neue Methode zur Integration der hyperelliptischen Differentialgleichungen und über die rationale Formihrer vollständigen algebraischen Integralgleichungen. J. Reine Angew. Math. 32, 220–227 (1846)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jacobi, C.G.J.: Vorlesungen über dynamik. G. Reimer, Berlin (1884)Google Scholar
  19. 19.
    Kiyohara, K.: Two-dimensional geodesic flows having first integrals of higher degree. Math. Ann. 320, 487–505 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kleiman S.L.: The Picard Scheme, Fundamental Algebraic Geometry. Mathematical Surveys and Monographs, vol. 123, pp. 235–321. American Mathematical Society, Providence, RI (2005)Google Scholar
  21. 21.
    Kuznetsov, V.B., Vanhaecke, P.: Bäcklund transformations for finite-dimensional integrable systems: a geometric approach. J. Geom. Phys. 44(1), 1–40 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Matveev, V.S., Topalov, P.J.: Integrability in the theory of geodesically equivalent metrics. J. Phys. A Math. Gen. 34, 2415–2434 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Miret, J.M., Moreno, R., Pujolàs, J., Rio, A.: Halving for the 2-Sylow subgroup of genus 2 curves over binary fields. Finite Fields Their Appl. 15(5), 569–579 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mumford, D.: Tata Lectures on Theta II. Birkhäuser, Boston (1984)Google Scholar
  25. 25.
    Sklyanin E.K.: Bäcklund transformations and Baxter’s Q-operator. In: Integrable Systems: from Classical to Quantum (1999, Montreal), CRM Proceedings. Lecture Notes, vol. 26, pp. 227–250. American Mathematical Society, Providence, RI (2000)Google Scholar
  26. 26.
    Sozonov, A.V., Tsiganov, A.V.: Bäcklund transformations relating different Hamilton-Jacobi equations. Theor. Math. Phys. 183, 768–781 (2015)CrossRefGoogle Scholar
  27. 27.
    Suris, Y.B.: The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, vol. 219. Birkhäuser, Basel (2003)CrossRefGoogle Scholar
  28. 28.
    Sutherland, A.V.: Fast Jacobian arithmetic for hyperelliptic curves of genus 3 (2016). arXiv:1607.08602
  29. 29.
    Tsiganov, A.V.: Canonical transformations of the extended phase space, Toda lattices and Stäckel systems. J. Phys. A Math. Gen. 33, 4169–4182 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tsiganov, A.V.: Killing tensors with nonvanishing Haantjes torsion and integrable systems. Reg. Chaotic Dyn. 20, 463–475 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tsiganov, A.V.: Simultaneous separation for the Neumann and Chaplygin systems. Reg. Chaotic Dyn. 20, 74–93 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Tsiganov, A.V.: On the Chaplygin system on the sphere with velocity dependent potential. J. Geom. Phys. 92, 94–99 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Tsiganov, A.V.: On auto and hetero Bäcklund transformations for the Hénon-Heiles systems. Phys. Lett. A 379, 2903–2907 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tsiganov A.V.: New bi-Hamiltonian systems on the plane (2017). arXiv:1701.05716MathSciNetCrossRefGoogle Scholar
  35. 35.
    Vanhaecke P.: Integrable Systems in the Realm of Algebraic Geometry. Lecture Notes in Mathematics, vol. 1638. Springer, Heidelberg (2001)Google Scholar
  36. 36.
    Weierstrass, K.: Über die geodätischen Linien auf dem dreiachsigen Ellipsoid. In: Mathematische Werke I, pp. 257–266, Berlin, Mayer and Müller (1895)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations