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

TL;DR

This paper develops techniques to identify a class of stable perturbations for minimal mass solitons in three-dimensional saturated nonlinear Schrödinger equations, demonstrating long-term persistence despite known nonlinear instability.

Contribution

It extends existing methods to establish stability of minimal mass solitons in 3D saturated NLS, showing their persistence under specific perturbations.

Findings

Existence of solutions with stable perturbations that persist over time.

Application of spectral projection and contraction mapping techniques.

Demonstration of long-term soliton persistence despite nonlinear instability.

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 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}.