NLS stability

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Suppose we are in the subcritical NLS , with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy . By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is -stable Ws1985, Ws1986. Below the 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 is betweeen and