The dynamics of the density of indirect excitons created in a double quantum well heterostructure by a laser illumination are considered and an idea of a generator of the traveling pulses of the excitonic condensed phase functioning at steady pumping is suggested. The setup of the proposed system includes a narrow highly illuminated region and a broad moderately illuminated region that supports the propagation of islands of the condensed phase. Simulation predicts that despite the steady pumping, the islands are formed periodically over time at the boundary of these regions and move due to the external potential bias. The calculations are based on the model of high density excitonic systems approbated by the explanations of multiple experiments. The generation of pulses occurs if both the uniform and nonuniform solutions for the exciton density exist in the propagation region. The range of the pumping intensity and the threshold for the potential bias for the generation are determined.