Singular Limits of the Klein–Gordon Equation

We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, , we show that the limit wave function of the modulated defocusing cub...

Full description

Saved in:
Bibliographic Details
Published in:Archive for rational mechanics and analysis 2010-08, Vol.197 (2), p.689-711
Main Authors: Lin, Chi-Kun, Wu, Kung-Chien
Format: Article
Language:English
Subjects:
Classical and quantum physics: mechanics and fields
Classical Mechanics
Complex Systems
Exact sciences and technology
Fluid- and Aerodynamics
Mathematical and Computational Physics
Physics
Physics and Astronomy
Quantum mechanics
Quantum statistical mechanics
Solutions of wave equations: bound states
Statistical physics, thermodynamics, and nonlinear dynamical systems
Theoretical
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, , we show that the limit wave function of the modulated defocusing cubic nonlinear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c → ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the defocusing nonlinear Schrödinger equation. The nonrelativistic-semiclassical limit, for some α  > 0, of the modulated defocusing cubic nonlinear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-010-0324-8