Isotropic curve flows

A smooth curve $\gamma$ in $\mathbb{R}^{n+1,n}$ is isotropic if $\gamma , \gamma_x, \dotsc , \gamma^{(2n)}_x$ are linearly independent and the span of $\gamma , \gamma_x, \dotsc , \gamma^{(n−1)}_x$ is isotropic. We construct two hierarchies of isotropic curve flows on $\mathbb{R}^{n+1,n}$, whose differential invariants are solutions of Drinfeld–Sokolov’s KdV type soliton hierarchies associated to the affine Kac–Moody algebra $\hat{B}^{(1)}_n$ and $\hat{A}^{(2)}_{2n}$. For example, the $\hat{B}^{(1)}_1$-KdV is the KdV hierarchy and the $\hat{A}^{(2)}_2$-KdV hierarchy is the Kupershmidt–Kaup (KK) hierarchy. Hence we our study gives geometric interpretations of the KdV and KK equations as the curvature flows of natural geometric curve flows on the light cone of $\mathbb{R}^{2,1}$. Bi-Hamiltonian structures and conservation laws for isotropic curve flows on $\mathbb{R}^{n+1,n}$ are also given.

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The authors’ research was supported in part by NSF of China Grant no. 11401327 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

Received 14 June 2018

Accepted 14 October 2019

Published 8 January 2021