Hamiltonian Curve Flows in Lie Groups G ⊂ U (n ) and Vector Nls, Mkdv, Sine-gordon Soliton Equations

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) ⊂ U(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 − 1)-invariant vector generalizations of both the nonlinear Schrödinger (NLS) equation and the complex modified Korteweg-de Vries (mKdV) equation, as well as U(N − 1)-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 Schrödinger maps, and mKdV analogs.