  Various kinds of Bose-Einstein condensates are considered, which evolve
without any geometric constraints or external trap potentials including
gravitational. For studies of their collective oscillations and stability,
including the metastability and macroscopic tunneling phenomena, both the
variational approach and the Vakhitov-Kolokolov criterion are employed,
calculations are done for condensates of an arbitrary spatial dimension. It is
determined that that the trapless condensate described by the logarithmic wave
equation is essentially stable, regardless of its dimensionality, while the
trapless condensates described by wave equations of a polynomial type with
respect to the wavefunction, such as the Gross-Pitaevskii (cubic),
cubic-quintic, and so on, are at best metastable. This means that trapless
"polynomial" condensates are unstable against spontaneous delocalization caused
by fluctuations of their width, density and energy, leading to a finite
lifetime.
