Квазиклассическое расширение квантовых газов в вакуум
In the framework of the Gross-Pitaevsky equation, we consider the problem of the expansion of quantum gases into vacuum, for which the chemical potential μ depends on the density n in a power manner with the index v =2/D, where D is the dimension of space. For gas condensates of Bose-atoms at temperatures T → 0 the main contribution to the interaction of atoms in the main order for the gas parameter is made by s-scattering, so for an arbitrary value of D, the index v =1. In the three-dimensional case, the value v =2/3 is realized for Fermi-atom condensates in the so-called unitary limit. The Gross-Pitaevsky equation when v = 2/D has additional symmetry with respect to Talanov transformations of conformal type, first found for a stationary self-focusing of light. A consequence of this symmetry is the virial theorem that connects the average size of the scattering gas cloud R and its Hamiltonian. Asymptotically, for t→∞, the value of R increases linearly with time. In the quasi-classical limit, the equations of motion coincide with the hydrodynamic equations of an ideal gas with an adiabatic index γ =1+2/D.
Self-similar solutions in this approximation are described against the background of an expanding angular deformations of a gas cloud within the framework of equations of Ermakov-Ray-Reid type.