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In this result, we develop the techniques of \cite{KS1} and \cite{BW} inorder to determine a class of stable perturbations for a minimal mass solitonsolution of a saturated, focusing nonlinear Schr\"odinger equation {c} i u_t +\Delta u + \beta (|u|^2) u = 0 u(0,x) = u_0 (x), in $\reals^3$. By projectinginto a subspace of the continuous spectrum of $\mathcal{H}$ as in \cite{S1},\cite{KS1}, we are able to use a contraction mapping similar to that from\cite{BW} in order to show that there exist solutions of the form e^{i\lambda_{\min} t} (R_{min} + e^{i \mathcal{H} t} \phi + w(x,t)), where $e^{i\mathcal{H} t} \phi + w(x,t)$ disperses as $t \to \infty$. Hence, we have longtime persistance of a soliton of minimal mass despite the fact that thesesolutions are shown to be nonlinearly unstable in \cite{CP1}.

A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schr\"odinger equations