Semiclassical Limit of the Gross-Pitaevskii Equation with Rotation

• Main Authors: Tsai, Jia-Ying, 蔡佳穎
• Other Authors: Lin, Chi-Kun
• Format: Others
• Language: en_US
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/79230680776356086085

碩士 === 國立交通大學 === 應用數學系所 === 99 === 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.