  A general form of $N$-dark soliton solutions of the multi-component Mel'nikov
system is presented. Taking the coupled Mel'nikov system comprised of
two-component short waves and one-component long wave as an example, its
general $N$-dark-dark soliton solutions in Gram determinant form are
constructed through the KP hierarchy reduction method. The dynamics of single
dark-dark soliton and two dark-dark solitons are discussed in detail. It can be
shown that the collisions of dark-dark solitons are elastic and energies of the
solitons in different components completely transmit through. In addition, the
dark-dark soliton bound states including both stationary and moving cases are
also investigated. An interesting feature for the coupled Mel'nikov system is
that the stationary dark-dark soliton bound states can exist for all possible
combinations of nonlinearity coefficients including all-positive, all-negative
and mixed types, while the moving case are possible when they take opposite
signs or they are both negative. The dynamics and several interesting
structures of the solutions are illustrated through some figures.
