Symmetry approach to the problem of the gas expansion into a vacuum

We present the results on the expansion of quantum and classical gases into a vacuum based on the use of symmetries. For quantum gases in the Gross–Pitaevsky (GP) approximation, additional symmetries arise for gases with a chemical potential $\mu$ that depends on the density $ n $ powerfully with exponent $ \nu = 2/D $, where $D$ is the space dimension. For gas condensates of Bose atoms at temperatures $T \to 0$, this symmetry arises for two-dimensional systems. For $D = 3$ and, accordingly, $\nu = 2/3$, this situation is realized for an interacting Fermi gas at low temperatures in the so-called unitary limit. The same symmetry for classical gases in three-dimensional geometry arises for gases with the adiabatic exponent $\gamma = 5/3$. Both of these facts were discovered in 1970 independently by V.I.Talanov for a two-dimensional nonlinear Schrödinger (NLS equation, which coincides with the Gross–Pitaevskii equation), describing stationary self-focusing of light in media with Kerr nonlinearity, and for classical gases, by S.I. Anisimov and Yu.I. Lysikov. In the quasiclassical limit, the GP equations coincide with the equations of the hydrodynamics of an ideal gas with the adiabatic exponent $\gamma = 1 + 2/D$. Self-similar solutions in this approximation describe the angular deformations of the gas cloud against the background of an expanding gas by means of Ermakov-type equations. Such changes in the shape of an expanding cloud are observed in numerous experiments both during the expansion of gas after exposure to powerful laser radiation, for example, on metal, and during the expansion of quantum gases into the vacuum.
The work of E.K. is supported by the Russian Science Foundation, Grant number 19-72-30028.