On Centro-Affine Curves and Bäcklund Transformations of the KdV Equation
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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.
KeywordsKorteweg-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.
- Arnold, M., Fuchs, D., Izmestiev, I., Tabachnikov, S.: Cross-ratio dynamics on ideal polygons. arXiv:1812.05337