NLS stability

Suppose we are in the $L^{2}\,$ subcritical NLS $p<1+2/d\,$, with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy $E\,$. By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is $H^{1}\,$-stable Ws1985, Ws1986. Below the $H^{1}\,$ norm, this is not known, but polynomial upper bounds on the instability are in CoKeStTkTa2003b. Multisolitons are also asymptotically stable under smooth decaying perturbations Ya1980, Grf1990, Zi1997, RoScgSf-p, RoScgSf-p2, provided that $p\,$ is betweeen $1+2/d\,$ and $1+4/d\,.$