具有電磁場的Klein-Gordon方程

Abstract

近年來關於一些著名的數學物理方程，例如Schrodinger方程，Ginzburg-Landau 方程的研究，逐漸移向探討具有電磁場的作用下方程式其解的行為。因此這個計畫主要是延續我們之前關於非線性 Klein-Gordon方程之研究。此時我們關心的是具有電磁場的非線性 Klein-Gordon方程。經由典型的Madelung變換或者所謂的WKB 方法，我們可以將之轉換為流體力學的形式。在這個形式底下，如果令光速趨近於無窮大，則極限方程式正是非線性 Schrodinger 方程式的流體形式；換句話說，電磁場在這個奇異極限下並沒有作用。因此想瞭解電磁場運作情形，我們必須對時間作適當的尺度變化(scaling),經由漸近分析我們發現有兩種極限情形，也就是長時間與更長時間；分別可得 anelastic (滯彈性)系統，與不可壓縮Euler方程。另外 仿Ginzburg-Landau方程，考慮具有電磁場的 Klein-Gordon方程，此時奇異極限是具有電磁場的波映射方程(wave map equation) 。

This Project is devoted to the study of the nonlinear Klein-Gordon equation with electromagnetic field. Formally letting ν→0, i.e., the nonrelativistic limit, we have the nonlinear Schrodinger equation with scalar potential $\phi$. It means that the magnetic vector potential $A$. As is well known in the semiclassical limit of the Schrodinger type equation, we have to introduce the Madelung transformation and transform the nonlinear Klein-Gordon equation with electromagnetic field to the hydrodynamics equations. From the relativistic quantum hydrodynamics equation, we have the quantum hydrodynamics equation as derived from the Schrodinger equation. To see the effect of the electromagnetic field, we have to rescale the time variable, then depending on the size of the scale, we have two different limit systems, one is the typical incompressible Euler equation, the other will be the anelastic approximation. Similar to the Ginzburg-Landau equation, we can consider the singular limit of the nonlinear Klein-Gordon equation with electromagnetic field. Employing the charge-energy inequality and the compactness technique, we can show that the limit equation will be the wave map equation with electromagnetic potential.