Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials
type: Item
instance of: scholarly article
title: Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials
zbMATH Open document ID: 0693.35132
DOI: 10.1007/BF01218621
author: Yong-Geun Oh
published in: Communications in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: This paper studies the Lyapunov stability of some semiclassical bound states found by \textit{A. Floer} and \textit{A. Weinstein} [J. Funct. Anal. 69, 397-408 (1986; Zbl 0613.35076)] and by the author [Commun. Partial Differ. Equations 13, No.12, 1499-1519 (1988)] of the nonlinear Schrödinger equation \[ i\hslash \partial \psi /\partial t=-(\hslash^ 2/2)\Delta \psi +V\psi -| \psi |^{p-1}\psi,\quad 1\leq p<1+4/n. \] It proves that among the bound states, those which are concentrated near local minima (respectively maxima) of the potential V are stable (respectively unstable). It also proves that those bound states are positive if \(\hslash >0\) is sufficiently small. More results on multi-lump bound states have been obtained also by the author [Commun. Math. Phys. (in press)].
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 4134653
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: semiclassical bound states
zbMATH Keywords: nonlinear Schrödinger equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4134653