  In this paper, a general bright-dark soliton solution in the form of Pfaffian
is constructed for an integrable semi-discrete vector NLS equation via Hirota's
bilinear method. One- and two-bright-dark soliton solutions are explicitly
presented for two-component semi-discrete NLS equation; two-bright-one-dark,
and one-bright-two-dark soliton solutions are also given explicitly for
three-component semi-discrete NLS equation. The asymptotic behavior is analysed
for two-soliton solutions.
