  Combining virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk,
Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear
induced dissipation on discrete modes, and in particular the notion of Refined
Profile, we show how to extend the theory by Kowalczyk, Martel, Munoz and Van
Den Bosch to the case when there is a large number of discrete modes in the
cubic NLS with a trapping potential which is associate to a repulsive potential
by a series of Darboux transformations. This a simpler model than the kink
stability for wave equations, but is still a classical one and retains some of
the main difficulties.
