Dynamical Behavior For The Blow-up Solutions Of Nonlinear Schr(?)dinger Equations With Potential

For studying of several problems in mathematical physics, the mathematical foundation of quantizing theory is an important subject, and nonlinear Schrodinger equation is a fundamental model in quantum mathematics mechanics. Classical nonlinear Schrodinger equation(without potential) is used to describe some phe-nomenons in quantum physics, such as the propagation of laser beam in dispersive and nonlinear medium , self trapping in nonlinear optics( [27])and Langmur waves in plasma( [74]). The nonlinear Schrodinger equation with potential has also definite physical background, especially the nonlinear Schrodinger equation with a harmonic potential is known as a model for describing the remarkable Bose-Einstein condensate(BEC) ( [7,8,68,69,78]).The nonlinear Schrodinger equation is a typical dispersive wave equation, which reflects the relation between dispersion and nonlinear interaction. When dispersion dominate, energy disperse in space and solution exists globally, decaying with time evolving( [13], [14], [23]). When the dispersion and nonlinearity reach balance, the onlinear Schrodinger equation has localized, finite energy solutions which are often standing waves( [3], [4], [17], [25], [26], [58], [60], [62], [78]). When the nonlinearity dominate, wave will collapse and the solution blows up in finite time( [64]).In the recent thirty years, a series of important advances are achieved on the mathematical studies on the nonlinear Schrodinger equation. Especially for...