In the present paper, we study the time-dependent correlation function of the one-dimensional impenetrable Bose gas, which can be expressed in terms of the Fredholm determinant of a time-dependent sine kernel and the solutions of the separated NLS equations. We derive the large time and distance asymptotic expansions of this determinant and the solutions of the separated NLS equations in both the space-like region and time-like region of the $(x,t)$-plane. Furthermore, we observe a phase transition between the asymptotic expansions in these two different regions. The phase transition is then shown to be described by a particular solution of the Painlev\'e IV equation.
Comment: 49 pages, 14 figures