The paper is devoted to the dynamics of dissipative gap solitons in the periodically corrugated optical waveguides whose spectrum of linear excitations contains a mode that can be referred to as a quasi-Bound State in the Continuum. These systems can support a large variety of stable bright and dark dissipative solitons that can interact with each other and with the inhomogeneities of the pump. One of the focus points of this work is the influence of slow variations of the pump on the behavior of the solitons. It is shown that for the fixed sets of parameters the effect of pump inhomogeneities on the solitons is not the same for the solitons of different kinds. The second main goal of the paper is systematic studies of the interaction between the solitons of the same or of different kinds. It is demonstrated that various scenarios of inter-soliton interactions can occur: the solitons can repulse each other or get attracted. In the latter case, the solitons can annihilate, fuse in a single soliton or form a new bound state depending on the kinds of the interacting solitons and on the system parameters.
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