I don't know what model sub you are working with, but many of them rely on using the wall to increase low frequency response. You don't want to eliminate that as you will kill the low frequencies.
I would look at several things. It looks like you may be hitting a resonant frequency. This can be solved by adding one or more cross braces to raise the resonant frequency. You could also add a second layer of sheetrock, or change the existing sheetrock for sound board. You might be able to get away with adding a thick padded wall covering. In general though, I think you will need to go into the wall and make structural changes. It might be cheaper to abandon the existing in wall unit and replace it with a conventional sub.
I definitely agree with charper about taking EVERYTHING off the wall and seeing if it goes away. It any be the stuff you hung up there and that can be fixed by mounting it more securely. I had a problem with some movie posters once, that I solved by tightening the mounting wire, getting the mounting hangers into studs and adding some foam tape to the bottom corners.
If all else fails, try isolating the sub from the wall. Does this sub have a rear enclosure, or is it open? If open, try sealing it off and add insulation inside. In any case, mount the sub to the frame with foam isolation. It will kill your bass, but greatly lower the coupling to the wall. One more thing. Is the sub mounted to a wood frame, or just attached to the plasterboard? If the later, you really should build a frame for the enclosure and attach this to the adjoining studs. This will help A LOT and may be all you need to do.
I'm sorry that in my opinion, any solution is probably going to require that you tear apart the wall to some degree.
