Hamiltonian curve flows in Lie groups $G\subset U(N)$ 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 $G=SO(N+1),SU(N)\subset U(N)$, 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 the nonlinear Schrodinger (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 Schrodinger maps, and mKdV analogs.