Instability of bound states for abstract nonlinear Schrödinger equations
type: Item
instance of: scholarly article
title: Instability of bound states for abstract nonlinear Schrödinger equations
zbMATH Open document ID: 1228.34092
author: Masahito Ohta
published in: Journal of Functional Analysis
publication date: 2011-05-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.1511
review text: The author studies abstract Hamiltonian systems of the form \[ u'(t) = \tilde{J}E'(u(t)), \] where \(E\) is the energy functional on a real Hilbert space \(X\), \(J\) is a skew symmetric operator on \(X\), and \(\tilde{J}\) is a natural extension of \(J\) to the dual space \(X^*\). One assumes that the above equation is invariant under a one parameter group \(\{T(s); \, s\in \mathbb{R}\}\) of unitary operators on \(X\). The instability of the bound states \(T(\omega t)\varphi_{\omega}\) is studied, where \(\omega \in \mathbb{R}\) and \(\varphi_{\omega}\) is a solution of the corresponding stationary problem. Applications to nonlinear Schrödinger equations are given.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 5898828
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: nonlinear Schrödinger equations
zbMATH Keywords: bound states
zbMATH Keywords: instability
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
OpenAlex ID: W1995073589
arXiv ID: 1010.1511
cites work: Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction
cites work: Instability of Standing Waves for a System of Nonlinear Schroedinger Equations with Three-Wave Interaction
cites work: Bifurcation from semitrivial standing waves and ground states for a system of nonlinear Schrödinger equations
cites work: Purely nonlinear instability of standing waves with minimal energy
cites work: Nonlinear bound states outside an insulated sphere
cites work: Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential
cites work: Nonlinear Schrödinger equation with a point defect
cites work: Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinger equations
cites work: Strong NLS soliton-defect interactions
cites work: Linearized instability for nonlinear schrödinger and klein-gordon equations
cites work: Stability theory of solitary waves in the presence of symmetry. I
cites work: Stability theory of solitary waves in the presence of symmetry. II
cites work: An instability mechanism for radially symmetric standing waves of a nonlinear Schrödinger equation
cites work: Symmetry-Breaking Bifurcation in Nonlinear Schrödinger/Gross–Pitaevskii Equations
cites work: Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential
cites work: Analysis.
cites work: Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity
cites work: Instability of bound states of nonlinear Schrödinger equations with Morse index equal to two
cites work: Strong Instability of Standing Waves for the Nonlinear Klein–Gordon Equation and the Klein–Gordon–Zakharov System
cites work: Instability of nonlinear bound states
cites work: Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation
cites work: Modulational Stability of Ground States of Nonlinear Schrödinger Equations
cites work: Lyapunov stability of ground states of nonlinear dispersive evolution equations
DOI: 10.1016/J.JFA.2011.03.010
description: scientific article