| This thesis studies the well-posedness and regularity of singular solutions of two classes of nonlinear quantum wave systems(Klein-Gordon-Schr(?)dinger systems and Maxwell-Schr(?)dinger systems).In Chapter 1,we give a brief overview of the background of these two systems and the current state of research in China and abroad,and introduce some necessary preliminary knowledge for this paper.In Chapter 2,we first study in detail the origin of the singular solutions of the Klein-Gordon-Schr(?)dinger system,and then prove the existence and regularity of the singular solutions of the system in the weaker topological space by using the Galerkin approximation.In particular,the uniqueness of the singular solution is proved by introducing an integral type of multiplier.Finally,the above theoretical analysis is supported by the time evolution figure from numerical simulations.It is shown that the space singular solution has similar properties to the original system.In Chapter 3,we extend the previous results by obtaining two singular solutions of the Maxwell-Schr(?)dinger system according to the Lorentz norm and verifying their well-posedness.The proof of its global existence is based on an a priori estimate obtained by deriving two conservation laws,L2-norm and energy,followed by evolutionary operators and normal transformations to prove the uniqueness of the solution.In fact,it can also be dealt with by an estimate related to the Schr(?)dinger operator with harmonic potential.And finally,the theoretical analysis is also verified using numerical simulations.In Chapter 4,we summarize the paper and give an overview of the future research work to be carried out. |
