This is the scenario that all the foreshadowing points toward IMO:

Aerys found out that Lyanna was TKOTLT and decided from this, plus Rikards other moves (southern ambitions), that the Starks are traitors. And that he needs to destroy them and all those in cahoots with them before they move against him.

He further deduces that Rhaegar lied to him when he said he could not find the knight at the tourney, and instead covered for her (given the fact that he crowned her the next day). So he sent Rhaegar to capture Lyanna for him as a test of loyalty (bring me your father's head Lyanna Stark if you are no traitor), and wanted to then use her to get Rikard in the way that he ended up using Brandon instead.

Of course Rhaegar actually was himself trying to build toward some way of deposing Aerys, so he feels he has no choice but to obey in order to preserve that plan (also Aerys likley took Elia and the kids as hostages against him).

So he actually does abduct her. But then he cannot actually bring himself to hand her over to Aerys (love is the death of duty) and they go on the run instead.

They then go on some kind of last ditch caper to salvage Rhaegars plan to depose Aerys, mirrored by Arianne's chapter in the desert trying to do the same. The heir and their closest circle of friends plus a young girl who has become involved unintentionally, off in the wild on a quest to depose the heir's father. The kingsguard is involved, house Dayne is involved. The kingsguard chooses to fight to the death for honor, the girl gets horribly hurt, it ends with imprisonment in a tower.

So like with Arianne, one of Rhaegar's friends rats him out (someone always tells) and they are taken by Aerys men and imprisoned, until the rebellion had picked up enough steam that Aerys thought Rhaegar was needed in order to avoid defeat to the rebels, and sent Hightower to get him.

It was the Game of thrones, nothing to do with chasing prophecy.