A Class Of Nonlinear Schr(?)dinger Equations In Bose-Einstein Condensates

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.