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Bäcklund Transformations and New Integrable Systems on the Plane

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Recent Developments in Integrable Systems and Related Topics of Mathematical Physics (MP 2016)

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Abstract

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.

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References

  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. 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)

    Article  MathSciNet  Google Scholar 

  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. Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comput. 48(177), 95–101 (1987)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  6. Dubrovin, B.A.: Riemann surfaces and nonlinear equations. Russ. Math. Surv. 36(2), 11–93 (1981)

    Article  Google Scholar 

  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. 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. Gantmacher, F.: Lectures in Analytical Mechanics. Mir Publishers, Moscow (1975)

    Google Scholar 

  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. 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. Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman and Hall/CRC, Boca Raton (2006)

    Google Scholar 

  13. Harley R.: Fast arithmetic on genus two curves (2000). http://cristal.inria.fr/~harley/hyper/

  14. Hietarinta, J.: Direct methods for the search of the second invariant. Phys. Rep. 147, 87–154 (1987)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  18. Jacobi, C.G.J.: Vorlesungen über dynamik. G. Reimer, Berlin (1884)

    Google Scholar 

  19. Kiyohara, K.: Two-dimensional geodesic flows having first integrals of higher degree. Math. Ann. 320, 487–505 (2001)

    Article  MathSciNet  Google Scholar 

  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. Kuznetsov, V.B., Vanhaecke, P.: Bäcklund transformations for finite-dimensional integrable systems: a geometric approach. J. Geom. Phys. 44(1), 1–40 (2002)

    Article  MathSciNet  Google Scholar 

  22. Matveev, V.S., Topalov, P.J.: Integrability in the theory of geodesically equivalent metrics. J. Phys. A Math. Gen. 34, 2415–2434 (2001)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  24. Mumford, D.: Tata Lectures on Theta II. Birkhäuser, Boston (1984)

    Google Scholar 

  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. Sozonov, A.V., Tsiganov, A.V.: Bäcklund transformations relating different Hamilton-Jacobi equations. Theor. Math. Phys. 183, 768–781 (2015)

    Article  Google Scholar 

  27. Suris, Y.B.: The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, vol. 219. Birkhäuser, Basel (2003)

    Book  Google Scholar 

  28. Sutherland, A.V.: Fast Jacobian arithmetic for hyperelliptic curves of genus 3 (2016). arXiv:1607.08602

  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)

    Article  MathSciNet  Google Scholar 

  30. Tsiganov, A.V.: Killing tensors with nonvanishing Haantjes torsion and integrable systems. Reg. Chaotic Dyn. 20, 463–475 (2015)

    Article  MathSciNet  Google Scholar 

  31. Tsiganov, A.V.: Simultaneous separation for the Neumann and Chaplygin systems. Reg. Chaotic Dyn. 20, 74–93 (2015)

    Article  MathSciNet  Google Scholar 

  32. Tsiganov, A.V.: On the Chaplygin system on the sphere with velocity dependent potential. J. Geom. Phys. 92, 94–99 (2015)

    Article  MathSciNet  Google Scholar 

  33. Tsiganov, A.V.: On auto and hetero Bäcklund transformations for the Hénon-Heiles systems. Phys. Lett. A 379, 2903–2907 (2015)

    Article  MathSciNet  Google Scholar 

  34. Tsiganov A.V.: New bi-Hamiltonian systems on the plane (2017). arXiv:1701.05716

    Article  MathSciNet  Google Scholar 

  35. Vanhaecke P.: Integrable Systems in the Realm of Algebraic Geometry. Lecture Notes in Mathematics, vol. 1638. Springer, Heidelberg (2001)

    Google Scholar 

  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 

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Authors and Affiliations

  1. St.Petersburg State University, St.Petersburg, Russia

    A. V. Tsiganov

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  1. A. V. Tsiganov
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Correspondence to A. V. Tsiganov .

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Editors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Victor M. Buchstaber

  2. Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia

    Sotiris Konstantinou-Rizos

  3. School of Mathematics, University of Leeds, Leeds, UK

    Alexander V. Mikhailov

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Tsiganov, A.V. (2018). Bäcklund Transformations and New Integrable Systems on the Plane. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_5

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