Will legendary Pokemon re-appear more than once?

Question

Ok so in most games you faint a legendary Pokemon then it comes back after I beat the elite 4 again, yea.
But just say it I crit it and whoops he is fainted again and I forgot to save. If I rematch the Elite 4 over again, will the legendary Pokemon come back?
In summary: will the legendary Pokemon re-appear every time I beat the Elite 4? (I'll give a scenario that would be needed).

Answer 1

Yes it will it will keep reappearing untill you catch it that is considering you keep beeting the pokemon leuge.