Integrable Systems in n-dimensional Riemannian Geometry
| dc.creator | Sanders, Jan A. | |
| dc.creator | Wang, Jing Ping | |
| dc.date | 2003-01-20 | |
| dc.date.accessioned | 2026-07-07T04:54:33Z | |
| dc.date.available | 2026-07-07T04:54:33Z | |
| dc.description | In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in (\orth{n+1}) with the vector modified Korteweg-De Vries equation (vmKDV) \[ \vk{t}= \vk{xxx}+\fr32 ||\vk{}||^2 \vk{x} \] as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the {\em natural} or {\em parallel} frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) {\em Fren{ê}t} frame. If one chooses the curvature to be zero, as is usual in the context of integrable systems, then one loses information unless one works in the natural frame. | |
| dc.identifier | https://arxiv.org/abs/math/0301212 | |
| dc.identifier | http://arxiv.org/abs/math/0301212 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66299 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Differential Geometry | |
| dc.subject | 35Q53, 53C21 | |
| dc.title | Integrable Systems in n-dimensional Riemannian Geometry | |
| dc.type | text |