# NLS stability

Suppose we are in the ${\displaystyle L^{2}\,}$ subcritical NLS ${\displaystyle p<1+2/d\,}$, with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy ${\displaystyle E\,}$. By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is ${\displaystyle H^{1}\,}$-stable Ws1985, Ws1986. Below the ${\displaystyle H^{1}\,}$ 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 ${\displaystyle p\,}$ is betweeen ${\displaystyle 1+2/d\,}$ and ${\displaystyle 1+4/d\,.}$