On Centro-Affine Curves and Bäcklund Transformations of the KdV Equation

  • Serge TabachnikovEmail author
Research Contribution


We continue the study of the Korteweg-de Vries equation in terms of cento-affine curves, initiated by Pinkall. A centro-affine curve is a closed parametric curve in the affine plane such that the determinant made by the position and the velocity vectors is identically one. The space of centro-affine curves is acted upon by the special linear group, and the quotient is identified with the space of Hill’s equations with periodic solutions. It is known that the space of centro-affine curves carries two pre-symplectic structures, and the KdV flow is identified with is a bi-Hamiltonian dynamical system therein. We introduce a one-parameter family of transformations on centro-affine curves, prove that they preserve both presymplectic structures, commute with the KdV flow, and share the integrals with it. Furthermore, the transformation commute with each other (Bianchi permutability). We also describe integrals of the KdV equation as arising from the monodromy of Riccati equations associated with centro-affine curves. We are motivated by our work (joint with M. Arnold, D. Fuchs, and I. Izmenstiev), concerning the cross-ratio dynamics on ideal polygons in the hyperbolic plane and hyperbolic space, whose continuous version is studied in the present paper.


Korteweg-de Vries equation Centro-affine curves Bäcklund transformation Riccati equation Bianchi permutability Bi-Hamiltonian structure 



It is a pleasure to acknowledge the stimulating discussions with A. Calini, A. Izosimov, I. Izmestiev, B. Khesin, and V. Ovsienko. Many thanks to the referee for useful comments and suggestions. This work was supported by NSF Grant DMS-1510055.


  1. Arnold, M., Fuchs, D., Izmestiev, I., Tabachnikov, S.: Cross-ratio dynamics on ideal polygons. arXiv:1812.05337
  2. Calini, A., Ivey, T., Marí-Beffa, G.: Remarks on KdV-type flows on star-shaped curves. Phys. D 238, 788–797 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Fujioka, A., Kurose, T.: Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves. Int. J. Geom. Methods Mod. Phys. 7, 165–175 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Fujioka, A., Kurose, T.: Multi-Hamiltonian structures on spaces of closed equicentroaffine plane curves associated to higher KdV flows. SIGMA 10, 11 (2014). Paper 048MathSciNetzbMATHGoogle Scholar
  5. Ince, E.L.: Ordinary Differential Equations. Dover Publications, New York (1944)zbMATHGoogle Scholar
  6. Khesin, B., Wendt, R.: The Geometry of Infinite-Dimensional Groups. Springer, Berlin (2009)zbMATHGoogle Scholar
  7. Ovsienko, V., Tabachnikov, S.: Projective Differential Geometry Old and New. From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  8. Ovsienko, V., Tabachnikov, S.: Coxeter’s frieze patterns and discretization of the Virasoro orbit. J. Geom. Phys. 87, 373–381 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Pinkall, U.: Hamiltonian flows on the space of star-shaped curves. Results Math. 27, 328–332 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Tabachnikov, S.: On the bicycle transformation and the filament equation: results and conjectures. J. Geom. Phys. 115, 116–123 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Terng, C.-L., Wu, Z.: Central affine curve flow on the plane. J. Fixed Point Theory Appl. 14, 375–396 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2019

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA

Personalised recommendations