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

Abstract

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.