The Geometric Airy Curve Flow on Rⁿ

2020/04/18 by Chuu-Lian Terng, Terng, Chuu-Lian
Mathematics · Physics and Astronomy · #53C44 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2004.08497

openalex publication_date 2020/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Langer and Perline proved that if x is a solution of the geometric Airy curve flow on Rⁿ then there exists a parallel normal frame along x(. ,t) for each t such that the corresponding principal curvatures satisfy the (n-1) component modified KdV (vmKdV_n). They also constructed higher order curve flows whose principal curvatures are solutions of the higher order flows in the vmKdV_n soliton hierarchy. In this paper, we write down a Poisson structure on the space of curves in Rⁿ parametrized by the arc-length, show that the geometric Airy curve flow is Hamiltonian, write down a sequence of commuting Hamiltonians, and construct Backlund transformations and explicit soliton solutions.

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