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A general form of $N$-dark soliton solutions of the multi-component Mel'nikovsystem is presented. Taking the coupled Mel'nikov system comprised oftwo-component short waves and one-component long wave as an example, itsgeneral $N$-dark-dark soliton solutions in Gram determinant form areconstructed through the KP hierarchy reduction method. The dynamics of singledark-dark soliton and two dark-dark solitons are discussed in detail. It can beshown that the collisions of dark-dark solitons are elastic and energies of thesolitons in different components completely transmit through. In addition, thedark-dark soliton bound states including both stationary and moving cases arealso investigated. An interesting feature for the coupled Mel'nikov system isthat the stationary dark-dark soliton bound states can exist for all possiblecombinations of nonlinearity coefficients including all-positive, all-negativeand mixed types, while the moving case are possible when they take oppositesigns or they are both negative. The dynamics and several interestingstructures of the solutions are illustrated through some figures.

General $N$-Dark Soliton Solutions of the Multi-Component Mel'nikov System