Dispersive Geometric Curve Flows

2014/11/08 by Terng, Chuu-Lian
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences

paper · doi:10.48550/arxiv.1411.2065

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 Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear 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 R³ and on R^2,1, the geometric Airy flow on Rⁿ, the Schrodingier 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.

Related

• Star Mean Curvature Flow on 3 manifolds and its Bäcklund Transformations
• Completely integrable curve flows on Adjoint orbits
• The Geometric Airy Curve Flow on Rⁿ
• Curve Flows and Solitonic Hierarchies Generated by (Semi) Riemannian Metrics
• Local existence of a fourth order dispersive curve flow on locally hermitian symmetric spaces and the application