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

In this result, we develop the techniques of \cite{KS1} and \cite{BW} in order to determine a class of stable perturbations for a minimal mass soliton solution 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 projecting into 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 long time persistance of a soliton of minimal mass despite the fact that these solutions are shown to be nonlinearly unstable in \cite{CP1}.