I don't think there is necessarily a problem, tbh - at least not as far as your DS is concerned, although clearly the situation is not ideal for you! My DS3 is similar right now - he is 17 weeks and although he is a lovely, easy baby in lots of ways (self-settles for the night at 7pm, naps well, very happy and smiley), he has recently been awake quite a bit in the wee small hours every night, kicking, fiddling with the cot bumper, singing (loudly) to himself etc. No crying, doesn't want a feed etc.
It is v tough, but I am trying to go with it for now. My understanding is that this is a pretty normal developmental stage (albeit not one that all babies experience in this way), and I certainly remember this being a tough age with DS1 and DS2 as far as sleeping at night was concerned. I am just trying not to do anything to reinforce the waking (e.g. no lights on, talking etc.) and hoping that it will pass soon - rather than seeing it as a problem that has to be 'solved', iyswim?
Arguably quite a passive approach, but I am v wary of imposing rigid routines and letting such small babies cry it out etc. - to me it feels like punishing them. I completely accept, however, that others have different views, and each to their own. Partly I think it depends on how you (and your baby) cope with the broken nights. I am quite lucky in that broken nights, while tough, don't leave me feeling like death warmed up, and DS3 seems just fine and dandy (and is, overall, getting enough sleep, judged on both his own demeanour and the guidance in all the books as to how much sleep a baby of this age should have). However, if I was hanging on by a thread with lack of sleep and/or my baby was clearly unhappy or extremely difficult to handle as a result of insufficient sleep, I daresay I'd be looking to take a more proactive approach.