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A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups G = SO(N + 1), SU (N) ⊂(N), generalizing previous work on integrable curve flows in Riemannian symmetric spaces G/SO(N). The derivation uses a parallel frame and connection along the curves, involving the Klein geometry of the group G. This is shown to yield the two known U(N – l)-invariant vector generalizations of both the nonlinear Schrodinger (NLS) equation and the complex modified Korteweg-de Vries (mKdV) equation, as well as U(N – l)-invariant vector generalizations of the sine-Gordon (SG) equation found in recent symmetry-integrability classifications of hyperbolic vector equations. The curve flows themselves are described in explicit form by chiral wave maps, chiral variants of Schrodinger maps, and mKdV analogs.

Hamiltonian Curve Flows in lie Groups G ⊂ U(N) and Vector Nls, mKdV, Sine-Gordon Soliton Equations