Integrable Flows for Starlike Curves in Centroaffine Space

Calini, Annalisa; Ivey, Thomas; Beffa, Gloria Maŕı

doi:10.3842/sigma.2013.022

Cited by 18 publications

(32 citation statements)

References 30 publications

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“…The Lax pair of the second flow in the A Higher order commuting central affine curve flows and a Poisson structure for (10.15) on R 3 \0 were given in [8]. 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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Section: Integrable Curve Flow On U With Constraintmentioning

confidence: 99%

Dispersive geometric curve flows

Terng

2014

Surveys in Differential Geometry

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“…Integrable curve flows in the centro-equiaffine geometry were discussed extensively in [21,24,33,35,40]. It turns out that the KdV equation arises naturally from a non-stretching curve flow in centro-equiaffine geometry.…”

Section: Bäcklund Transformations Of the Kdv And Camassa-holm Flowsmentioning

confidence: 99%

“…Motions of curves in the affine geometry were discussed in [13,21,23,33,40]. It is well-known that the Sawada-Kotera equation arises from a non-stretching curve flow in affine geometry.…”

Section: Bäcklund Transformations Of the Sawada-kotera Flowmentioning

confidence: 99%

Bäcklund Transformations for Integrable Geometric Curve Flows

Han

Kang

2015

Symmetry

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Abstract:We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kotera flow in the affine geometry, etc. Using the fact that two different curves in a given geometry are governed by the same integrable equation, we obtain Bäcklund transformations relating to these two integrable geometric flows. Some special solutions of the integrable systems are used to obtain the explicit Bäcklund transformations.

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“…x ) the central affine moving frame, and u i the i-th central affine curvature of γ for 1 ≤ i ≤ n − 1 (cf. [3], [8]). Note that…”

Section: Introductionmentioning

confidence: 99%