A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schrödinger equations

Abstract

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 Schrödinger equation {c} i u_t + Δu + β(|u|^2) u = 0 u(0,x) = u_0 (x), in ${\mathbb{R}}^3$. 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 λ_{\min} t} (R_{min} + e^{i \mathcal{H} t} ϕ+ w(x,t)), where $e^{i\mathcal{H}t}\varphi +w(x,t)$ 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}.

22 pages, 1 figure

References in corpus (3)

• Fast soliton scattering by delta impurities
• Asymptotic stability of N-soliton states of NLS
• Stable manifolds for an orbitally unstable NLS