The dissertation focuses on the nonlinear Schrodinger equation iu_t+u_{xx}+kappa|u|^2u =0, for the complex-valued function u=u(x,t) with domain t>=0, 0<=x<= L, where the parameter kappa is any non-zero real number. It is shown that the problem is locally and globally well-posed for appropriate initial data and the solution exponentially decays to zero as t goes to infinity under the boundary conditions u(0,t) = beta u(L,t) and beta u_x(0,t)-u_x(L,t) = ialpha u(0,t), where L>0, and alpha and beta are any real numbers satisfying alpha*beta<0 and beta does not equal 1 or -1.
Moreover, the numerical study of controllability problem for the nonlinear Schrodinger equations is given. It is proved that the finite-difference scheme for the linear Schrodinger equation is uniformly boundary controllable and the boundary controls converge as the step sizes approach to zero. It is then shown that the discrete version of the nonlinear case is boundary null-controllable by applying the fixed point method. From the new results, some open questions are presented. / Ph. D. / The dissertation concerns the solutions of nonlinear Schrodinger (NLS) equation, which arises in many applications of physics and applied mathematics and models the propagation of light waves in fiber optics cables, surface water-waves, Langmuir waves in a hot plasma, oceanic and optical rogue waves, etc. Under certain dissipative boundary conditions, it is shown that for given initial data, the solutions of NLS equation always exist for a finite time, and for small initial data, the solutions exist for all the time and decay exponentially to zero as time goes to infinity. Moreover, by applying a boundary control at one end of the boundary, it is shown using a finite-difference approximation scheme that the linear Schrodinger equation is uniformly controllable. It is proved using fixed point method that the discrete version of the NLS equation is also boundary controllable. The results obtained may be applicable to design boundary controls to eliminate unwanted waves generated by noises as well as create the wave propagation that is important in applications.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/77506 |
| Date | 24 April 2017 |
| Creators | Cui, Jing |
| Contributors | Mathematics, Sun, Shu-Ming, Lin, Tao, Kim, Jong U., Yue, Pengtao |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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