We investigate formation of Bose-Einstein condensates under non-equilibrium conditions using numerical simulations the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and forcing at high wave numbers, study propagation turbulent spectrum toward low numbers. Our primary goal is to compare results for evolving with previous obtained kineti...

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the one-step (abelian) and the two-step ones, there are only six (here f(m, p) ...

We classify solvable Lie groups with free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the one-step (abelian) and the two-step ones, there are only six others: (here f(...

We demonstrate the existence of phase fluctuations in elongated Bose-Einstein Condensates (BECs) and study the dependence of those fluctuations on the system parameters. A strong dependence on temperature , atom number, and trapping geometry is observed. Phase fluctuations directly affect the coherence properties of BECs. In particular, we observe instances where the phase coherence length is s...

We examine the occurrence of Bose-Einstein condensation in both nonrelativistic and relativistic systems with no self-interactions in a general setting. A simple condition for the occurrence of Bose-Einstein condensation is given. We show that condensation can occur only if q ≥ 3, where q is the dimension associated with the continuous part of the eigenvalue spectrum of the Hamiltonian for nonr...