  In this paper, we firstly establish the multi-Hamiltonian structure and
infinite many conservation laws for the vector Kaup-Newell hierarchy of the
positive and negative orders. The first nontrivial negative flow corresponds to
a coupled Fokas-Lenells equation. By constructing a generalized Darboux
transformation and using a limiting process, all kinds of one-soliton solutions
are constructed including the bright-dark soliton, the dark-anti-dark soliton
and the breather-like solutions. Furthermore, multi-bright and multi-dark
soliton solutions are derived and their asymptotic behaviors are investigated.
