HAMILTONIAN FORMALISM FOR THE HIGHER KdV FLOWS ON THE SPACE OF CLOSED COMPLEX EQUICENTROAFFINE CURVES

Fujioka, Atsushi; Kurose, Takashi

doi:10.1142/s0219887810003987

Cited by 14 publications

(20 citation 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%

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%

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

Tabachnikov

2018

Arnold Math J.

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“…There have been several works on the curve flow (10.13) on R 2 \0 concerning the higher flows and the bi-Hamiltonian structure (cf. [7], [15], [16], [35]). Wu and the author also constructed Bäcklund transformations and proved the periodic Cauchy problem for (10.13) has long time existence in [35].…”

Section: Integrable Curve Flow On U With Constraintmentioning

confidence: 99%

Dispersive geometric curve flows

Terng

2014

Surveys in Differential Geometry

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Abstract. The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrödinger flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural nonlinear dispersive curve flows in geometric analysis. A curve flow is integrable if the evolution equation of the local differential invariants of a solution of the curve flow is a soliton equation. For example, the Hodge star mean curvature flow on R 3 and on R 2,1 , the geometric Airy flow on R n , the Schrödinger flow on compact Hermitian symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a compact Lie group are integrable. In this paper, we give a survey of these results, describe a systematic method to construct integrable curve flows from Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the Cauchy problem of these curve flows.

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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%