How do all the automatics cope with a ski jump?
No-one knows*. It's not done one yet. However, I can't think of a good reason why it shouldn't be easy and safe**. I spent a lot of time looking at how another digital active-control V/STOL jet would do it. This was done in order to design a monitoring system that would allow us to experiment with experimental control laws trying to cope with finding themselves suddenly in the air without any
immediate "right answer" to the question of what to do with the engine, nozzles, flaps and pitch-controls to achieve at least 1g flight. The trick then was to make sure the system knew it was doing a ski-jump, without anyone having to tell it (they might forget to do that). That was not difficult, because there are some big unambiguous clues that are easy to reliability detect. Having detected it, having the FCS do the right thing achieve 1+g flight in a timely manner is pretty easy (it would be basically like having the right take-off trim setting and fading it to "ordinary flying" appropriately after ramp-exit. I do not know exactly what the F-35B does, but that sort of thing worked for us).
I suppose you could worry about steering, which old-school catapulted jets have done for them, but even that's easier on a ski-jump than flat deck because the visual cues are so much better and there's no need to raise the nose during the T/O roll.
So I imagine it will do it very nicely.
*But they must be, rightly in my opinion, pretty confident.
**noting of course that, in the words of the guidance notes for a once-popular DStan: "nothing is safe".