【作者】 魏赟赟；

【导师】 郭科； 马永旺；

【作者基本信息】 成都理工大学 ， 应用数学， 2007， 硕士

【摘要】 以2001年诺贝尔物理学奖为标志，Bose—Einstein凝聚的研究已成为当今国际物理学界研究的几个热点领域之一．我们将根据描述Bose-Einstein凝聚的一类数学模型，一类带调和势并具组合幂非线性项的非线性Schrodinger方程，对Bose-Einstein凝聚所特别关注的如下问题进行研究：1该方程的整体解和爆破解存在的条件及最佳条件；2该方程的驻波解的性态．整个论文的方法是现代变分法．通过分析这个方程的特征，以方程的Cauchy问题的局部适定性为基础，构造合适的泛函和Nehari流形，从而设置相应的强制变分问题，通过分析这些变分问题的特性，构造某些特定的函数．结合这些函数特征、方程的特征以及一些重要不等式的特征，建立了它的所谓发展不变流．然后，讨论该方程的整体解存在性与有限时间内的爆破性质．最后结合变分特征确定出该方程整体解存在的最佳条件．进一步，讨论了其驻波解的存在性和不稳定性．第一章，介绍了方程的相关物理背景、已有研究状况、问题，以及这篇论文的工作．第二章，运用变分方法，研究了方程整体解和爆破解存在的条件．从理论上获得了该方程Cauchy问题的整体解和爆破解的一个分界门槛．第三章，用能量泛函作为判别准则，给出了方程整体解和爆破解的门槛条件．同时，回答了在能量空间中，初值到底要小到什么程度，其解才会整体存在?特别值得一提的是，本章的结论是可以用于实际计算的，因此它更有实际应用价值．第四章，研究了方程的驻波解．证明了该方程驻波解的存在性，进一步，证明了其驻波的不稳定性．更多还原

【Abstract】 The Nobel physics prize in 2001 indicated that the research about Bose-Einstein Condensates is very important in the modern international physics. According to one of the mathematical models in Bose-Einstein Condensates, a nonlinear Schr(?)dinger equation with a harmonic potential and combined power type nonlinearities, we shall research the following problems in Bose-Einstein Condensates.1. The conditions of global existence, blowup and sharp threshold.2. The existence and properties of the standing waves.Our method is the modern variational method. Firstly, we analyze the characteristics of the equation basing on the local well-posedness of the Cauchy problem of the equation. Then we set some proper functionals and Nehari manifolds to pose constrained variational problem. Combining the characteristics of the equation with the variational problem and some important inequalities, we construct some evolution invariant flows of the equation. After that, we study the global existence and blowup properties. Then according to the variational problem, we derive out the sharp criteria of blowup and global existence of the equation. At last, we obtain the existence and instability of the standing waves.In the first chapter, we show the physical background and some known research results of the nonlinear Schr(?)dinger equation. At the same time, we also present our goal and the main results of the paper.In the second chapter, using the variational method, we consider the conditions of global existence and blowup. Furthermore, we obtain the sharp threshold between the global existence and blowup of the Cauchy problem of the equation.In the third chapter, using the energy functional as the criterion, we get another sharp threshold of global existence and blowup of the solutions of the equation. Furthermore, we answer the following problem: how small are the initial data such that the global solution exist? It is deserved to note that the results in this chapter can be calculated in the applications of Bose-Einstein Condensates.In the last chapter, we are interested in the standing waves of the equation. We firstly prove the existence of the standing waves. Then we obtain that the instability of the standing waves.更多还原