A Class of Stable Perturbations for a Minimal Mass Soliton in Three-Dimensional Saturated Nonlinear Schrödinger Equations

DOI10.1137/09075175XzbMATH Open1210.35236arXiv0906.0375MaRDI QIDQ5392377

Publication date: 8 April 2011

Published in: SIAM Journal on Mathematical Analysis (Search for Journal in Brave)

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 + �eta (|u|^2) u = 0 u(0,x) = u_0 (x), in

e a l s^{3}

. By projecting into a subspace of the continuous spectrum of

m a t h c a l 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 m a t h c a l H t} p h i + w (x, t)

disperses as

t o i n f t y

. 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}.

Full work available at URL: https://arxiv.org/abs/0906.0375

zbMATH Keywords

Schrödinger equations soliton stability

Mathematics Subject Classification ID

Stability in context of PDEs (35B35) NLS equations (nonlinear Schrödinger equations) (35Q55) Soliton equations (35Q51) Perturbations in context of PDEs (35B20) Soliton solutions (35C08)