The Research Summary Of The Properties Of The Blow-up Solutions For A Nonlinear Schr(?)dinger Equation

Nonlinear Schrodinger equations are fundamental equations of the quantum mechanics , which originate from the quantum field theory, specially Hartree-Fock theory. From the mathematical point of view, the nonlinear Schrodinger equation possesses a mixture of the similar properties of parabolic and hyperbolic equations. In recent years,the nonlinear Schrodinger equation received a great deal of attention from mathematicians. Not only because it has promulgated the basic rule of the motion of matter in microphysics world, moreover it has widely applications in the nonlinear optics domain .Indeed,some simplified models lead to certain nonlinear Schrodinger equations.This article has done some research summary on the property of the blow-up solution for a nonlinear Schrodinger equation. First, the article discusses systematically the solution of the nonlinear Schrodinger equation will blow up in the case of finite variance,the case of infinite variance,and in non-isotropic spaces respectively. And take a quasilinear Schrodinger equation for example ,not only discussing the blow up problem of its solution,moreover,promoting predecessor's result,and proving that the solution of the quasilinear Schrodinger equation will blow up in nonisotropic spaces. Then,The article analyses the property of the blow-up solution for a nonlinear Schrodinger equation in detail. In the end, here Proposes several questions that has not yet solved at present.