Unitarily-invariant integrable systems and geometric curve flows in SU(n+1)/U(n) and SO(2n)/U(n)

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 modfied Korteveg-de Vries (mKdV) equation,as well as a novel nonlocal multi-component version of the nonlinear Schrodinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flowsgiven by a non-stretching wave map and a mKdV analog of a non-stretching Schrodinger 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.