非線性Klein-Gordon方程的色散極限

Abstract

本論文主要研究非線性Klein-Gordon方程的色散極限問題。首先，我們從Klein-Gordon 方程嚴格數學的推導到可壓縮與不可壓縮的歐拉方程。在極限系統出現奇異點前，非相對論-半古典極限可推導到可壓縮的歐拉方程。假如我們考慮時間的尺度變換，則半古典極限(光速固定)可以得到不可壓縮的歐拉方程。 我們也完成了有關非線性Klein-Gordon方程的奇異極限問題，包含了半古典極限、非相對論極限與非相對論-半古典極限。有關半古典極限，我們證明了三次非線性的Klein-Gordon方程其波函數收斂到有相對論效應的wave map方程，且對應的相函數滿足有相對論效應的線性波方程。另外，非相對論極限的非線性Klein-Gordon方程收斂到非線性的薛丁格方程。最後，有關非相對論-半古典極限，我們證明了三次非線性的Klein-Gordon方程其波函數收斂到wave map方程，且對應的相函數滿足線性波方程。

This dissertation investigates the dispersive limits of the nonlinear Klein-Gordon equations. First, we perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein-Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we further rescale the time variable, then in the semiclassical limit (the light speed kept fixed), the incompressible Euler equations are recovered. We also 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 of the modulated defocusing nonlinear Klein-Gordon equation is the defocusing nonlinear Schrodinger equation. The nonrelativistic-semiclassical limit of the modulated defocusing cubic nonlinear Klein-Gordon equation is the classical wave map for the limit wave function and typical linear wave equation for the associated phase function.