有旋轉項的Gross-Pitaevskii方程之半古典極限

Abstract

在本論文中，我們用兩種不同的做法研究有旋轉項的Gross-Pitaevskii方程之半古典極限。首先，我們使用修改過的Madelung變換以著重在與量子流體動力學方程（quantum hydrodynamical equations）等價的擬線性雙曲對稱系統（quasilinear symmetric hyperbolic system）。我們建立在極限系統奇點形成之前，當普朗克常數趨近於零時，量子密度與量子動量收斂到可壓縮的旋轉歐拉方程（compressible rotational Euler equation）之唯一解。此外，我們證明在維度2之可壓縮的旋轉歐拉方程之局部解的存在性與唯一性。其次，我們考慮量子密度與量子動量在恆定狀態(1,0)附近的情形。我們建立有旋轉項的Gross-Pitaevskii 方程弱收斂到等價於線性波動方程（linear wave equation）的波映射方程（wave map equation）。這方法的結果引領聲波（acoustic wave）的討論。

In this paper, we perform the semiclassical limit of the Gross-Pitaevskii equation with rotation by two different approaches. First, we use the modified Madelung transformation to focus on the quasilinear symmetric hyperbolic system, which is equivalent to the quantum hydrodynamical equations. We establish that before the formation of singularities in the limiting system, the quantum density and quantum momentum converge to the unique solution of the compressible rotational Euler equation as the Planck constant ħ tends to zero. In addition, we prove the existence and uniqueness of local solutions of the compressible rotational Euler equation in dimension 2. Second, we consider the case when the quantum density and quantum momentum are near the constant state (1,0). We establish that the Gross-Pitaevskii equation with rotation converges weakly to the wave map equation, equivalently the linear wave equation. The result of this approach leads the discussion of the acoustic wave.