Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

Fujioka, Atsushi; Kurose, Takashi

doi:10.3842/sigma.2014.048

Cited by 9 publications

(13 citation statements)

References 34 publications

(47 reference statements)

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“…Starting with U. Pinkall [9], a number of recent papers were devoted to the study of the Korteweg-de Vries equation in terms of cento-affine curves [2,3,4,11]. Let us present the relevant results.…”

Section: A Family Of Transformations On the Space Of Curvesmentioning

confidence: 99%

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

Tabachnikov

2018

Arnold Math J.

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A family of transformations on the space of curvesThis note stems from [1] where we study the integrable dynamics of a 1parameter family of correspondences on ideal polygons in the hyperbolic plane and hyperbolic space: two n-gons P = (p 1 , p 2 , . . .) and Q = (q 1 , q 2 , . . .)In the limit n → ∞, a polygon becomes a parameterized curve. The ground field can be either R of C; to fix ideas, choose R. Let us use the following definition of cross-ratio to define our correspondence (other five definitions result in the change of the constant c):We replace polygons by non-degenerate closed curves γ : R → RP 1 with γ ′ (t) > 0; to be concrete, let the period be π: γ(t + π) = γ(t). Also let us assume that the rotation number of the curve γ is 1, that is, γ : R/πZ → RP 1 is a diffeomorphism. Denote the space of such curves by C and let C = C/ PSL(2, R) be the moduli space.Then a continuous version of (1) is γ ′ (t)δ ′ (t) (δ(t) − γ(t)) 2 = c.

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Section: A Family Of Transformations On the Space Of Curvesmentioning

confidence: 99%

“…It is shown in [2,11,4] that the forms ω and Ω provide a bi-Hamiltonian structure on C, corresponding to a pair of compatible Poisson brackets for the KdV equation.…”

Section: A Family Of Transformations On the Space Of Curvesmentioning

confidence: 99%

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

Tabachnikov

2018

Arnold Math J.

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“…(2) Ker((L 3 ) q ) is the space of all ξ ∈ C ∞ (S 1 , R) such that ξ(γ) = Aγ for some A ∈ sl(2, R). Next we write down the formula for the Poisson operators induced from L 2j+1 via the map Ψ: Note that 1 4ŵ 5 is the symplectic form (1.5) defined by Pinkall in [15], andŵ 3 is the symplectic form defined by Fujioka and Kurose in [9]. (e) The order (2j + 1) central affine curve flow (2.15) is the Hamiltonian flow forĤ 2(j−i)+1 with respect to the Poisson structureĴ 2i+1 for all 0 ≤ i ≤ j.…”

Section: Bi-hamiltonian Structurementioning

confidence: 99%

“…Higher order central affine curve flows and conservation laws for (1.3) were given in [5], [3], [8]. A bi-Hamiltonian structure for (1.3) was discussed in [9].…”

Section: Introductionmentioning

confidence: 99%

Central affine curve flow on the plane

Terng

2013

J. Fixed Point Theory Appl.

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Abstract. We give the following results for Pinkall's central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a biHamiltonian structure, (ii) Bäcklund transformations and a permutability formula, (iii) infinitely many families of explicit solutions. We also solve the Cauchy problem for periodic initial data.

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“…When n = 2 and 3, it is possible to explicitly compute the conditions on the (coefficients of the) operator P , and one finds that the curvature u(t) must satisfy an equation of the n-KdV hierarchy ( [25,11,12,5,6,14]), but this becomes unfeasible for general n. Hence, to determine the analogous conditions for general n, we solve the same problem for the spectral-parameterized linear ODE L(ψ) = λψ in § 2.6, using an approach reminiscent of [8,13,2].…”

Section: Introductionmentioning

confidence: 99%