Unitarily-invariant integrable systems and geometric curve flows in $SU(n+1)/U(n)$ and $SO(2n)/U(n)$
Abstract
Bi-Hamiltonian hierarchies of soliton equations are derived from geometric non-stretching (inelastic) curve flows in the Hermitian symmetric spaces and . The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schr\"odinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flows given by a non-stretching wave map and a mKdV analog of a non-stretching Schr\"odinger map in the case of the SG and mKdV systems, and a generalization of the vortex filament bi-normal equation in the case of the NLS systems.
Keywords
Cite
@article{arxiv.1408.5290,
title = {Unitarily-invariant integrable systems and geometric curve flows in $SU(n+1)/U(n)$ and $SO(2n)/U(n)$},
author = {Ahmed M. G. Ahmed and Stephen C. Anco and Esmaeel Asadi},
journal= {arXiv preprint arXiv:1408.5290},
year = {2018}
}
Comments
32 pages. Typos fixed. To appear in J. Phys. A