Ahh. I've experimented heaps with this problem. I reckon I've solved it. You knobs who reckon you don't even tie it on, that's bullshit. You must never stand up and must have seats the size of armchairs. It's not an Adventure if you never stand up.
I cut loops out of old bike tyres, about 10mm wide so that they are strong enough not to break, but not too unstretchy. I use four loops, one attaches to each corner of the airhawk. Then I use two bits of twine (or any light thin rope would do) to tie two loops to each other around under the seat. i.e. the two rear to each other, the two front to each other.
The tyre loops (Kind of like super strong rubber bands) sit against the seat edges and the corner underneath and the rubber/vinyl friction stops them moving backwards and forwards. You only see the black tyre loops against the (usually) black seat as the twine is only under the seat.
Takes a bit of fiddling getting the twine in the right place and the right tension, but works great. There is still some movement and flex in the rubber bands/tyre loops for you to slightly move the airhawk around if you need to, and to stop the Airhawk's loops tearing if you drag your leg across it, crash etc.