Spectral Theory and Nonlinear PDE: a Survey

2005/09/01 by Wilhelm Schlag, Schlag, Wilhelm · 2 citations
Engineering · Mathematics · Physics and Astronomy · #35B35 #35B40 #35Q51 #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations #math-ph #math.AP #math.MP #msc:35B35 #msc:35B40 #msc:35Q51 #msc:35Q55

paper · pdf · doi:10.48550/arxiv.math/0509019

openalex publication_date 2005/09/01 · arxiv created 2005/09/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a slightly modified version of the survey article. Minor changes to the text and the references have been made.

Citations

• Spectral properties of Schrödinger operators and time-decay of the wave functions
• Numerical verification of a gap condition for linearized NLS
• Asymptotic stability of N-soliton states of NLS
• Stable manifolds for all monic supercritical NLS in one dimension
• Solitary Wave Dynamics in an External Potential
• Uniqueness of positive solutions of Δu−u+up=0 in Rn
• Orbital stability of standing waves for some nonlinear Schr�dinger equations
• Time decay for solutions of Schr�dinger equations with rough and time-dependent potentials
• Existence of solitary waves in higher dimensions
• A remark on Littlewood-Paley theory for the distorted Fourier transform
• Stable Directions for Excited States of Nonlinear Schrödinger Equations

Cited by

• Why are solitons stable?
• Why are solitons stable?

Related

• Spectral Methods in PDE
• Introduction to Nonlinear Spectral Analysis
• Spectral theory of dynamical systems
• Spectral stability and spatial dynamics in partial differential equations
• What are symmetries of nonlinear PDEs and what are they themselves?