  Consider a branch of unstable solitons of NLS whose linearized operators have
one pair of simple real eigenvalues in addition to the zero eigenvalue. Under
radial symmetry and standard assumptions, solutions to initial data from a
neighbourhood of the branch either converge to a soliton, or exit a larger
neighbourhood of the branch transversally. The qualitative dynamic near a
branch of unstable solitons is irrespective of whether blowup eventually
occurs, which has practical implications for the description of blowup of NLS
with supercritical nonlinearity.
