ANALOG GRAVITY IN GROSS-PITAEVSKII EQUATION

We study the dynamics of a Bose-Einstein condensate (BEC) to find solutions that correspond to a sonic black/white hole. In such configurations, the condensate goes from subsonic to supersonic when crossing a particular region that can be understood as a horizon. This is because sound cannot go back from the supersonic to the subsonic region. Therefore, in this set-up the speed of sound plays the role same as that of the speed of light in a gravitational black/white hole, an important difference being that there are excitations that can go faster than the speed of sound and therefore can escape (enter) the sonic black (white) hole. These analog gravity models are of a particular interest in the study of the nature of black holes in simple laboratory setups. Here, the motion of the Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation (GPE). We discuss singular Stationary [1] and Self-similar [2] solutions of Gross-Pitaevskii Equation (GPE) in 2D (with Circular symmetry) and 3D (with Spherical symmetry). We use these solutions to study the approximate local speed of sound and magnitude of flow velocity of the condensate to see whether they cross, indicating the potential existence of a sonic analog of a black/white hole. We discuss different approaches to solve for these solutions, such as numerical techniques and the semi-analytical Laplace-Borel resummation of asymptotic transseries solutions. We check these Laplace-Borel resummations of the transseries to see how well they agree with numerical solutions. Particularly in the case of stationary solutions, we also study and demonstrate the utility of deep learning for these differential equations, phenomenon of resurgence in transseries, and trans-asymptotic summation. Furthermore, considering the practical limitations in achieving the singularities in the stationary condensate density, we also regularize [3] (remove the singularity from) the singular stationary solutions by putting a background spatial metric and/or artificially constructed specific kinds of external potentials and variable couplings so that this configuration can (at least as a better approximation) be achieved in a laboratory setup. We use these to study fluctuations in the density and phase of the regularized solutions and derive acoustic metrics under some approximations to study these analog gravity systems in context of Hawking radiation (temperature) through some approximations.