My jetlag experiences have all been with eastbound vacation trips from US mainly to England, but also Turkey, various European cities etc. With my earplugs and eyemask, and a determination to convince myself that I CAN sleep, I do usually -- maybe not well, or soundly, but enough that I am surprised by the arrival of the breakfast cart. Trips to England have been frequent enough lately that I've learned to adjust, and it seems my body can handle that change fine. Coming home (westbound) is often harder.
I think it's the excitement of vacation and the adrenaline that carry me through most of the first day -- though I might doze off if I'm on a train for any length of time.
On a longer trip, I did try Melatonin. It worked for getting to sleep, but every time I've used it I have very vivid and disrupting dreams (so I don't do that any more). Benadryl kind of stuff makes me wired. I guess I'll just have to deal.
When I talked to my dr. about it, he did offer an Rx. He added the caveat that he was once on a longhaul trip where many pax took sleep aids on wheels-up -- and when the plane had to turn back for some reason and land an hour after takeoff, they were all sound asleep on landing. Took a while to shovel them out of their seats and into the terminal...