  We consider the possible existence of gravitationally bound general
relativistic strings consisting of Bose-Einstein condensate (BEC) matter which
is described, in the Newtonian limit, by the zero temperature time-dependent
nonlinear Schr\"odinger equation (the Gross-Pitaevskii equation), with
repulsive interparticle interactions. In the Madelung representation of the
wave function, the quantum dynamics of the condensate can be formulated in
terms of the classical continuity equation and the hydrodynamic Euler
equations. In the case of a condensate with quartic nonlinearity, the
condensates can be described as a gas with two pressure terms, the interaction
pressure, which is proportional to the square of the matter density, and the
quantum pressure, which is without any classical analogue though, when the
number of particles in the system is high enough, the latter may be neglected.
Assuming cylindrical symmetry, we analyze the physical properties of the BEC
strings in both the interaction pressure and quantum pressure dominated limits,
by numerically integrating the gravitational field equations. In this way we
obtain a large class of stable stringlike astrophysical objects, whose basic
parameters (mass density and radius) depend sensitively on the mass and
scattering length of the condensate particle, as well as on the quantum
pressure of the Bose-Einstein gas.
