  By examining the bound-state spectra of transverse fluctuations about
one-dimensional, spatially localized dark and bright soliton wavetrains of the
Gross-Pitaevskii equation, it is established that the low-temperature ground
states of repulsive and attractive quasi-one-dimensional Bose-Einstein
condensates are degenerate. In the one-soliton limit, both ground states are
shown to possess two distinct transverse fluctuation modes which can couple to
the spatial soliton: the first involves zero-energy exchange, costing only the
soliton shape dressing via uniform translation of its centre of mass. The
second mode contributes in a negative energy for the repulsive case, but
positive energy in the attractive case. This unstability of the repulsive
spatial soliton against localized transverse fluctuation modes invalidates the
Gross-Pitaevskii equation for stationary solitonic processes in repulsive 1D
Bose-Einstein-Condensate systems.
