Cross-Invariant Manifolds For The Gross-Pitaevskii-Poisson Equation And Their Applications

In 1995,E.A.Cornell,C.E.Wieman observed a new state of matter—Bose-Einstein condensation in the laboratory,which greatly promotes the development of quantum mechanics and related research fields.In order to further study Bose-Einstein condensation,Gross and Pitaevskii established the famous Gross-Pitaevskii equation by using the mean field method.The mathematical model describing dipole Bose-Einstein condensation is the Gross-Pitaevskii-Poisson equation.In this paper,the two-dimensional Gross-Pitaevskii-Poisson equation will be investigated by using the variational method.By constructing the cross-invariant manifolds,the criterion for the existence of blowup solutions and the strong instability of standing waves will be studied.The specific contents are as follows:In Chapter 1,the research background,main methods and main results of this paper are introduced.In Chapter 2,the two-dimensional Gross-Pitaevskii-Poisson equation with potential is researched.The modified cross-constrained variational problems are defined and the modified cross-invariant manifolds are constructed.Then sharp threshold for the existence of blowup solution and global solution is obtained.Further more,the strong instability of standing waves is obtained.In Chapter 3,the two-dimensional Gross-Pitaevskii-Poisson equation without potential is studied.By defining new cross-constrained variational problems,the cross-invariant manifolds are obtained.Then the criterion for the existence of blowup solution and the strong instability of standing waves are obtained.