  We consider the kick-induced mobility of two-dimensional (2D) fundamental
dissipative solitons in models of lasing media based on the 2D complex
Ginzburg-Landau (CGL) equation including a spatially periodic potential
(transverse grating). The depinning threshold is identified by means of
systematic simulations, and described by means of an analytical approximation,
depending on the orientation of the kick. Various pattern-formation scenarios
are found above the threshold. Most typically, the soliton, hopping between
potential cells, leaves arrayed patterns of different sizes in its wake. In the
laser cavity, this effect may be used as a mechanism for selective pattern
formation controlled by the tilt of the seed beam. Freely moving solitons
feature two distinct values of the established velocity. Elastic and inelastic
collisions between free solitons and pinned arrayed patterns are studied too.
