|1- Ahmed, A., Anco, S. C., Asadi, E., "Unitarily-invariant integrable systems and geometric curve flows in SU (n + 1)/U(n) and SO(2n)/U(n)", Journal of Physics A: Mathematical and Theoretical, 51: (6), 065205-1 -065205-35, (2018).|
Bi-Hamiltonian hierarchies of soliton equations are derived from geometric non-stretching (inelastic) curve flows in the Hermitian symmetric spaces $SU(n+1)/U(n)$ and $SO(2n)/U(n)$ . 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 Schrödinger (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 Schrödinger 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.