The Star Trek warp drive has some features in common with the Alcubierre drive, a theoretical faster-than-light drive proposed by physicist Miguel Alcubierre in 1994. There is no canonical statement by producers or writers of the series to suggest that the warp drive is the Alcubierre drive.
From the Wikipedia article on the Alcubierre drive:
The Star Trek television series used the term "warp drive" to describe their method of faster-than-light travel. Neither the Alcubierre theory, nor anything similar, existed when the series was conceived, but Alcubierre stated in an email to William Shatner that his theory was directly inspired by the term used in the show, and references it in his 1994 paper.
In particular, the Alcubierre drive uses a "warp bubble".
From the same article:
The ship would then ride this wave inside a region of flat space, known as a warp bubble, and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive.
As it stands, the Alcubierre metric is the best model we currently have for understanding Star Trek's warp drive at a mathematical / general relativistic level. (Caution: being the best model does not mean it is a good model).