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A two-dimensional (2D) generalization of the stabilized Kuramoto -Sivashinsky (KS) system is presented. It is based on the Kadomtsev-Petviashvili(KP) equation including dissipation of the generic (Newell -- Whitehead --Segel, NWS) type and gain. The system directly applies to the description ofgravity-capillary waves on the surface of a liquid layer flowing down aninclined plane, with a surfactant diffusing along the layer's surface.Actually, the model is quite general, offering a simple way to stabilizenonlinear waves in media combining the weakly-2D dispersion of the KP type withgain and NWS dissipation. Parallel to this, another model is introduced, whosedissipative terms are isotropic, rather than of the NWS type. Both modelsinclude an additional linear equation of the advection-diffusion type, linearlycoupled to the main KP-NWS equation. The extra equation provides for stabilityof the zero background in the system, opening a way to the existence of stablelocalized pulses. The consideration is focused on the case when the dispersivepart of the system of the KP-I type, admitting the existence of 2D localizedpulses. Treating the dissipation and gain as small perturbations and making useof the balance equation for the field momentum, we find that the equilibriumbetween the gain and losses may select two 2D solitons, from their continuousfamily existing in the conservative counterpart of the model (the latter familyis found in an exact analytical form). The selected soliton with the largeramplitude is expected to be stable. Direct simulations completely corroboratethe analytical predictions.Comment: a latex text file and 16 eps files with figures; Physical Review E,  in pres

Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations