Defect self-propulsion in active nematic films with spatially-varying activity

We study the self-propulsion of topological defects in an active nematic system with spatially varying activity, and consider the two limits of a constant activity gradient, and a sharp jump in the activity. For an activity with a constant gradient the self-propulsion velocity of the +1/2 defect becomes proportional to the local activity in the same way it is for a constant activity. A finite vorticity is induced at the defects core, which drives the defect to align its polarization with the direction of decreasing absolute activity, so that it will start moving in the direction that lowers its self-propulsion velocity. The constant activity gradient does not make the -1/2 defect motile and it does not pick up a preferred orientation. A sharp jump in the activity affects the translational and rotational motion of both defects. The positive defect slow down when it approaches the interface and an induced vorticity tends to reorient it so that the defects polarization becomes perpendicular to the interface. The -1/2 defect becomes self-propelled close to the wall and a finite vorticity is induced that tends to align the defects in a preferred orientation. The negative defects that has the prefered orientation is attracted by the interface.