On instability of some approximate periodic solutions for the full nonlinear Schr\"odinger equation

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.