论文标题
具有更高维度的诱捕电势的非线性schrödinger方程
Nonlinear Schrödinger equations with trapping potentials in higher dimensions
论文作者
论文摘要
从数学方面，通常通过变异方法研究非线性schrödinger方程，从而停止在较高维度上工作。本论文试图通过关注球形对称的解决方案来克服这一问题。然后，可以使用来自普通微分方程和动态系统领域的经典方法。此处介绍的结果包括固定溶液的存在和独特性，它们的频率和稳定性。还探讨了谐振近似的动力学特性。主要重点是Schrödinger-Newton-Hooke方程，该方程式被证明是反DE保姆时空扰动的非偏差限制。
From the mathematical side, nonlinear Schrödinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric solutions. Then, one can use classical methods coming from the fields of ordinary differential equations and dynamical systems. The results presented here include existence and uniqueness of the stationary solutions, their frequency, and stability. The dynamical properties of the resonant approximation are also explored. The main focus is given to the Schrödinger-Newton-Hooke equations that is shown to be a nonrelativistic limit of perturbations of the anti-de Sitter spacetime.
