The classic story concerns a brilliant physicist who figures out - mathematically - how to slip back in time. But she succumbs to Alzheimer's before she can convert her equations into actual results. Fortunately (or unfortunately) her husband is a brilliant engineer who is able to apply his wife's equations. His passion for this is not so much for his work, but to get his wife back to a time, four years earlier, when she still had all of her faculties.
As is always the case in these stories, the time slips have adverse effects beyond the time slipping couple. The time displacement in each shift backwards endangers people miles away, when the rug of time is pulled out from under them. The worst case is yet to come - hundreds of people will be trapped and killed in a tunnel, which didn't exist four years ago.
Peter gets called in when Olivia thinks his presence may have caused the time slips. By the end of the episode, our team knows it's the engineer - but Peter thinks he was the cause, after all, because the equipment built by the engineer didn't start working until Peter arrived in this universe three days ago (that's three weeks ago in television episode time).
And the ending proceeds to be double sad. The physicist, in her final sojourn in brilliant mentality, crosses out all of her equations, rather than providing the last piece of mathematics which could make the time shift permanent, as the husband so desperately wanted.
And Peter apparently comes to believe that he really doesn't belong here.
But I'll bet you as far as Peter is concerned that Olivia and Walter will come to take and bring him back.
... Fringe 4.2: Better and Worse Selves ... Fringe 4.3: Sanity and Son ... Fringe 4.4: Peter's Back, Ectoplasm, and McLuhan ... Fringe 4.5: Double Return