Self-similar configurations in Bose–Einstein condensates, their Borel analysis, and acoustic horizons

Abstract Self-similarity simplifies the analysis of nonlinear differential equations governing complex systems such as quantum superfluids. Self-similar solutions can capture universal features of singularities and sonic horizons in analog gravity systems. In this article, we study self-similar configurations of non-relativistic Bose–Einstein condensates described by the Gross–Pitaevskii equation. Specifically, we examine singular self-similar solutions in two spatial dimensions with circular symmetry and in three spatial dimensions with spherical symmetry. These solutions allow us to identify transitions between subsonic and supersonic condensate flow, leading to the emergence of supersonic regions and analog sonic horizons, providing a self-similar sonic analog of a black hole. In such configurations, phonons cannot propagate from the supersonic region to the subsonic region against the condensate flow, and the transition boundary defines the sonic horizon. We further study semi-analytical Laplace-Borel resummation of asymptotic series and associated transseries expansions to construct approximate solutions. Numerical methods are employed to validate the semi-analytical results and to analyze the solution structure.