Central affine curve flow on the plane

Terng, Chuu-Lian; Wu, Zhiwen

doi:10.1007/s11784-014-0161-8

Cited by 10 publications

(11 citation statements)

References 22 publications

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“…This is indeed the case: (11) implies that 1 µ + 1 ν = 1, and then a calculation shows that 1 µ γ − µγ 2 , γγ 2 (µ − 1) 1 − µ, γµ − γ 2…”

Section: Thus We Need To Show Thatmentioning

confidence: 83%

“…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%

“…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%

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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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“…This is indeed the case: (11) implies that 1 µ + 1 ν = 1, and then a calculation shows that 1 µ γ − µγ 2 , γγ 2 (µ − 1) 1 − µ, γµ − γ 2…”

Section: Thus We Need To Show Thatmentioning

confidence: 83%

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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“…(c) Analogues of the Hasimoto transforms were also constructed between the Schrödinger flow on Gr(k, C n ) and the matrix NLS in [33], between central affine curve flows on R n \{0} and the GelfandDickey hierarchy in [23] and [7] for n = 2, in [8] for n = 3, and in [36] for n ≥ 3, and between the shape operator curve flow on Adjoint orbits of U (n) on u(n) and the n-wave equation in [13] and [28]. We will give a brief survey of these results in the last section.…”

Section: H(γ(· T))mentioning

confidence: 99%

“…A bi-Hamiltonian structure, higher order curve flows, and Bäcklund transformations for (10.15) on R n \0 are given and the periodic Cauchy problem for (10.15) on R n \0 is solved in [36].…”

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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Integrable systems and invariant curve flows in symplectic Grassmannian space

Song

Yao

2017

Physica D: Nonlinear Phenomena

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Central affine curve flow on the plane

Cited by 10 publications

References 22 publications

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

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

Dispersive geometric curve flows

Integrable systems and invariant curve flows in symplectic Grassmannian space

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