  We consider a Bose-Einstein condensate (BEC) with attractive two-body
interactions in a cigar-shaped trap, initially prepared in its ground state for
a given negative scattering length, which is quenched to a larger absolute
value of the scattering length. Using the mean-field approximation, we compute
numerically, for an experimentally relevant range of aspect ratios and initial
strengths of the coupling, two critical values of quench: one corresponds to
the weakest attraction strength the quench to which causes the system to
collapse before completing even a single return from the narrow configuration
("perihelion") in its breathing cycle. The other is a similar critical point
for the occurrence of collapse before completing two returns. In the latter
case, we also compute the limiting value, as we keep increasing the strength of
the post-quench attraction towards its critical value, of the time interval
between the first two perihelia. We also use a Gaussian variational model to
estimate the critical quenched attraction strength below which the system is
stable against the collapse for long times. These time intervals and critical
attraction strengths---apart from being fundamental properties of nonlinear
dynamics of self-attractive BECs---may provide clues to the design of upcoming
experiments that are trying to create robust BEC breathers.
