Bäcklund Transformations for Integrable Geometric Curve Flows

Abstract

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.

On Backlund transformations of surfaces by extended Harry-Dym flow

“…In this paper, we choose that both curve flows for γ and γ are governed by the same integrable system, that means the curvatures of the curves γ determined by the flow (4) satisfy the integrable systems as for the curves [24].…”

“…Let us recall the Extended version of Harry-Dym flow given by Qu [24]. The extended version of Harry-Dym flow:…”

Visco-Da Rios Equation in 3-Dimensional Riemannian Manifold

“…Also, they described Bäcklund transformation of a null Cartan curve in Minkowski 3-space as a transformation which maps a null Cartan helix to another null Cartan helix. Qu, Han and Kang [13] investigated Bäcklund transformations relating to binormal flow and extended Harry-Dym flow as integrable geometric flows. Some special solutions of the integrable systems are used to obtain the explicit Bäcklund transformations.…”

Local well-posedness and blow-up for a family of U(1)-invariant peakon equations

“…In addition, the CH equation has several interesting geometrical aspects. It describes geodesic flows in the volume-preserving diffeomorphism group of the line (or circle) [16,28,29,41], as well as non-stretching curve flows in centro-equiaffine planar geometry [38]. Moreover, it possesses algebro-geometric solutions on a symplectic submanifold [36].…”