An Essay On Some Problems Of Nonautonomous Stochastic Schr(?)dinger Equations And Fractional Differential Systems

The thesis studies several kinds of non-autonomous stochastic nonlinear Schrodinger equations under the background of nonlinear optics, mainly including the local and global well-posedness as well as the limit behavior of solutions. Meanwhile, the monotone iterative solutions for a class of nonlinear fractional differential systems with deviating arguments are also considered. The present Ph.D. thesis is divided into seven chapters.Chapter 1 introduces the historical backgrounds and current situations of stochas-tic Schrodinger equation and fractional differential equation, and presents the main works of our thesis.Chapter 2 aims to outline some preliminaries on deterministic and stochastic Schrodinger equation, and to sketch some element notions and important conclusions in probability and stochastic analysis and fractional differential equation.Chapter 3 is devoted to a random Schrodinger equation with time-oscillating non-linearity and dissipation/gain. Firstly, the existence of local solution for the stochastic Schrodinger equation with white noise dispersion and the averages of time-oscillating terms is established in H1(Rd). Then the existence and local convergence of the solu-tion for the original equation is proved. Our results generalize the partial conclusions of related papers and improve some known results.Chapter 4 is intended to consider the limit behavior of a random Schrodinger e-quation with time-dependent linear loss/gain and time-periodic dispersion. Firstly, we modify the stochastic Strichartz type estimates established in some related paper. Then, we use these Strichartz type estimates to prove the local existence of solution for the Schrodinger equation with white noise dispersion and the average of time-periodic dis-persion. Lastly, we prove the convergence of the solution for the original equation to the solution of the Schrodinger equation with white noise dispersion. The results improve some known research conclusions.Chapter 5 is aimed to study a random Schrodinger equation with nonlinear time-dependent loss/gain. Firstly, we consider the global well-posedness of the stochastic Schrodinger equation with white noise dispersion and nonlinear loss/gain in L2(Rd) and H1(Rd). Then, under some assumptions, we prove the existence and global con-vergence of solution for the original equation. Our results are the generalization and improvement of the conclusions of some papers.Chapter 6 is concerned with the monotone iterative solutions of the nonlinear frac-tional order differential systems with deviating arguments. Firstly, two well-defined monotone sequences are introduced and proved to converge uniformly to the solution-s of the systems. Meanwhile, a numerical iterative scheme is introduced to obtain accurate approximate solutions for the systems. In addition, we apply a numerical ex-periment to demonstrate the accuracy and efficiency of the new approach. These results improve the conclusions of related article.In the final Chapter 7, some summaries on our works and future studies are given out.