ON SMOOTHNESS AND UNIQUENESS OF MULTI-SOLITONS OF THE NON-LINEAR SCHRÖDINGER EQUATIONS

In this paper, we study some properties of multi-solitons for the non-linear Schrodinger equations in R^d with general non-linearities. Multi-solitons have already been constructed in H^1, successively by Merle, by Martel and Merle, and by Cote, Martel and Merle. We show here that multi-solitons are smooth, depending on the regularity of the non-linearity. We obtain also a result of uniqueness in some class, either when the ground states are all stable, or in the mass-critical case.