Diffusion bound and reducibility for discrete Schrödinger equations with tangent potential
type: Item
instance of: scholarly article
title: Diffusion bound and reducibility for discrete Schrödinger equations with tangent potential
zbMATH Open document ID: 1268.37086
author: Shiwen Zhang
author: Zhiyan Zhao
published in: Frontiers of Mathematics in China
publication date: 2013-04-10T00:00:00Z
review text: The lattice Schrödinger equations with a spatial localized real bounded (tangent) potential are considered. The diffusion for a class of such equations is studied in detail. This problem falls within the same general category of bounds on the higher Sobolev norms (\(H^1\) or beyond) for the continuum nonlinear Hamiltonian partial differential equation in a compact domain. It is proven that the growth of the \(H^1\) norm of the solution of the given lattice Schrödinger equations is at most logarithmic for some fixed initial data. Furthermore, it is supposed that the linear equation has a time quasi-periodic potential. Then it is proven that the linear equation can be reduced to an autonomous equation for the most values of the frequency. The Birkhoff normal form is used to study the dynamics of the respective Hamiltonian system described by the considered lattice Schrödinger equations.
Mathematics Subject Classification ID: 37K55
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 37K60
zbMATH DE Number: 6152752
zbMATH Keywords: tangent potential
zbMATH Keywords: reducibility
zbMATH Keywords: Sobolev norm
zbMATH Keywords: Birkhoff normal form
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
full work available at URL: s11464-012-0241-2
OpenAlex ID: W2025551880
cites work: Localization in \(\nu\)-dimensional incommensurate structures
cites work: Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential
cites work: On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential
cites work: Anderson localization for time quasi-periodic random Schrödinger and wave equations
cites work: Q5430043
cites work: Polynomially decaying transmission for the nonlinear Schrödinger equation in a random medium
cites work: On reducibility of Schrödinger equations with quasiperiodic in time potentials
cites work: Localization in disordered, nonlinear dynamical systems
cites work: Quasi-periodic breathers in Hamiltonian networks of long-range coupling
cites work: Quasi-periodic Solutions for One-Dimensional Nonlinear Lattice Schrödinger Equation with Tangent Potential
cites work: Q5429849
cites work: Growth of Sobolev norms and controllability of the Schrödinger equation
cites work: Anderson Localization for Time Periodic Random Schrödinger Operators
cites work: Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations
cites work: Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations
cites work: Long time Anderson localization for the nonlinear random Schrödinger equation
cites work: Q3333453
cites work: Perturbations of lower dimensional tori for Hamiltonian systems
DOI: 10.1007/S11464-012-0241-2
description: scientific article; zbMATH DE number 6152752