  Using the Fermi Golden Rule analysis developed in several results by the
first author, we prove asymptotic stability of asymmetric nonlinear bound
states bifurcating from linear bound states for a quintic nonlinear
Schr\"odinger operator with symmetric potential. This goes in the direction of
proving that the approximate periodic solutions of the NLS shadowed in a recent
work of the second author with Michael Weinstein do not persist for the full
NLS.
