  We consider the generalized Good-Boussinesq model in one dimension, with
power nonlinearity and data in the energy space $H^1\times L^2$. This model has
solitary waves with speeds $-1<c<1$. When $|c|$ approaches 1, Bona and Sachs
showed orbital stability of such waves. It is well-known from a work of Liu
that for small speeds solitary waves are unstable. In this paper we consider in
more detail the long time behavior of zero speed solitary waves, or standing
waves. By using virial identities, in the spirit of Kowalczyk, Martel and
Mu\~noz, we construct and characterize a manifold of even-odd initial data
around the standing wave for which there is asymptotic stability in the energy
space.
