The Dynamic Property Of The Nonlinear Schr(?)dinger Equation And The Theoretical Study Of Bose-Einstein Condensation

The nonlinear Schr(o|¨)dinger equation as a ubiquitous nonlinear evolution equation plays an important role in nonlinear physics, and it has a wide application in many areas of physics, including laser fusion, plasma physics, nonlinear optics and Bose-Einstein Condensation. In recent years, many researchers have been paid more attention to the nonlinear dynamics, and discovered many interesting phenomena. In value research aspect, the week inter-particle interactions and the diluteness of the gases allow for a mean-field description of the system of BEC, and consequently granted the widely use of the mathematical model, the Gross-Pitaevskii(GP)equation, and the form of this equation is similar to nonlinear Schr(o|¨)dinger equation.Bose-Einstein Condensate (BEC) is the Bose gas below a certain transition temperature, a finite fraction of the total number of the particles would occupy the lowest-energy single-particle state. The BEC is realized in experiment since 1995. This has stimulated many theoretical studies on the properties of condensates. Many researchers were interested in BEC, and had lots of articles to make the comprehensive discussion from each aspect to the BEC phenomenon.Nonlinear Schr(o|¨)dinger equation is an infinite-dimensional Hamiltonian system. But the Hamiltonian system canonical equation in sysmplectic transformation under the form is invariable. The time-evolution of Hamiltonian system is the evolution of sysmplectic transformation. Therefore, Ruth and Feng Kang presented the symplectic algorithm for solving the Hamiltonian system. Symplectic algorithm is the difference method that preserves the symplectic structure, and it is a better method in the calculation of long-time many-step and preserving the structure of system.In this thesis we investigate the dynamic properties of one-dimensional cubic nonlinear Schr(o|¨)dinger equation and drifting of the solution pattern by using the symplectic algorithm with different nonlinear parameters in the perturbation initial condition. In the numerical calculations, we first discretize the spatial derivative of the nonlinear Schr(o|¨)dinger equation: substitute the space difference approximation for 6 orders the spatial derivative, transform the nonlinear Schr(o|¨)dinger equation to Hamiltonian canonical equations, and then get the discrete Hamiltonian. For the other method, we expand the wave function with the B-spline. We find the discrete Hamiltonian, and then transform the nonlinear Schr(o|¨)dinger equation to Hamiltonian canonical equations. Hamiltonian system has a symplectic structure. So we solve the equation with implicit symplectic scheme of 2-step 4-order Runge-Kutta. The numerical result demonstration, the nonlinear parameter increases from 0.01 to 0.85, the phase trajectories of in the phase space by the elliptic orbit would be similar to the homoclinic orbit (HMO). The phase trajectories turns to the accurate cyclical circulation by the precise cycle circulating motion, and the phase trajectories gradually increases in the horizontal axis value. With the nonlinear parameter increasing, drifting velocity of the solution pattern becomes faster at the same time of evolution.This article takes example for 1D time-dependent GP equation, which numerically simulates interaction of two BECs and three BECs. We numerically solve 1D time-dependent GP equation by using the symplectic algorithm, and transform the nonlinear Schr(o|¨)dinger equation to Hamiltonian canonical equations. With the procedure proposed by Ruprecht et al, we tackled the equation by Euler-center algorithm which is also a symplectic algorithm. The interaction of BECs is investigated when the trapping potential exists and the trapping potential is set to zero. The result indicates that the elastic collision will occur between the two and three BECs when the trapping potential exists. If the trapping potential is set to zero at t=0, the interference fringe is observed between two and three BECs, and with nonlinear parameter increasing, condensates interference fringe gets stronger at the same time. And three BECs interference are divided into two processes. While if the trapping potential is set to zero at t=2, the interaction between two and three BECs is complex.