The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential
The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential
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DOI：
10.1137/040614554
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
N. Abdallah;F. Méhats;C. Schmeiser;Rada-Maria Weishäupl
中科院分区：
文献类型：
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作者：
N. Abdallah;F. Méhats;C. Schmeiser;Rada-Maria Weishäupl
研究了具有多项式增长势和简谐禁闭势一般非线性的非线性薛定谔方程。更准确地说，禁闭势是强各向异性的；即不同方向上的陷阱频率具有不同的数量级。当陷阱频率比趋于零时，进行了限制。在主谐振子基态上的质量集中被传播，并且低维调制波函数再次满足非线性薛定谔方程。分析的主要工具是能量和Strichartz估计，以及两个各向异性的索博列夫不等式。作为应用，讨论了模拟玻色-爱因斯坦凝聚动力学的三维Gross-Pitaevskii方程的降维问题。
The nonlinear Schrodinger equation with general nonlinearity of polynomial growth and harmonic confining potential is considered. More precisely, the confining potential is strongly anisotropic; i.e., the trap frequencies in different directions are of different orders of magnitude. The limit as the ratio of trap frequencies tends to zero is carried out. A concentration of mass on the ground state of the dominating harmonic oscillator is shown to be propagated, and the lower-dimensional modulation wave function again satisfies a nonlinear Schrodinger equation. The main tools of the analysis are energy and Strichartz estimates, as well as two anisotropic Sobolev inequalities. As an application, the dimension reduction of the three-dimensional Gross-Pitaevskii equation is discussed, which models the dynamics of Bose-Einstein condensates.