Qualitative Analysis For A System Of Nonlinear Schr(o|¨)dinger Equation With Potential

Schr(o|¨)dinger equation was put forward by Austrian physicist Schr¨odinger it is thebasic equation in quantum mechanicsis. Schr(o|¨)dinger equation reveals the motion lawof microscopic particles and very important for the research of quantum mechanicsis.On the other hand, Schr(o|¨)dinger equation is a typical model of dispersion equation.It can be also applied in the field of Physics, Chemistry and Optics. Therefore, it isvery interesting to study the properties for the solution, meanwhile, new directions andchallenges are proposed in the field of mathematical physics equations.However, it is often technically simpler than other dispersive equation like thewave equation or Kdv equation. By using energy method and Strichartz estimate,mathematicians study both the problems of local nature (local existence of solutions,uniqueness, regularity, smoothing impact) and the problems of global nature(globalexistence, finite-time blowup, asymptotic behavior of solutions, scattering). So far, aperfect theoretical structure is established.The paper mainly studies the initial value problems of the nonlinear Schrodingerequations with potentials in n-d Euclidean space, as follows We care for blow-up and global existence of equation solution. Our main con-tents are summary and development of some previous works directly. By establishing atype of cross constrained variational problem and constructing so-called cross-invariantmanifolds, We can get a sharp threshold for blow up and global existence of the solution.