Dynamical Study Of The Interaction Between Bose-Einstein Condensates

The nonlinear Schr?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 paid more attention to the nonlinear dynamics, and discovered many interesting phenomena. The week inter-particle interactions and the diluteness of the gases allow for a mean-field description of the system of BEC, and consequently grant the widely use of the mathematical model, the Gross-Pitaevskii (GP) equation, and the form of this equation is similar to the nonlinear Schr?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. BEC is realized in experiment since 1995. This has stimulated many theoretical studies on the properties of the condensates. Many researchers were interested in BEC, and had written lots of articles to make comprehensive discussions from each aspect to the BEC phenomenon.Nonlinear Schr?dinger equation is an infinite-dimensional Hamiltonian system. But the Hamiltonian system canonical equation is invariable under symplectic transformation. The time-evolution of Hamiltonian system is the evolution of symplectic 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.What is unique in our computation is that the symplectic algorithm is employed since the GP equation can be discretized into a Hamiltonian system which has symplectic structure. Symplectic algorithm is the difference method that preserves the symplectic structure, and is a better method in the calculation of long-time many-step and preserving the structure of system. The fundamental theorem of Hamiltonian mechanics says that the time-evolution of Hamiltonian system is the evolution of symplectic transformation. In this sense, the Hamiltonian system has a symplectic structure. Therefore, the symplectic algorithm is appropriate for solving the Hamiltonian system, and for solving the Hamiltonian mechanics.This thesis mainly consists of three specific topics in the field of numerically study of BEC: (i) Fundamental study of the ground state wavefunction of BEC in spherically harmonic trap by time-independent method. (ii) Dynamical study of the numerical solution of BEC by the time-dependent method. (iii) Numerical study of the interference of two condensates. The main parts of the thesis are summarized here:The results show that the time evolution of cigar-shaped condensates is the same as that of a soliton when the trapping potential exists. Supposed there exits the harmonic oscillator potential, the ground state wave function will diffuse very slowly with the non-linear coefficient increasing, on the contrary, without the amendment term, the wave packet expands more quickly. At the same time, the time evolution of the particle density is not fixed in certain point, but a regular alternate change of particle density. We call this phenomenon "Breathing". Moreover, the greater the nonlinear parameter is, the smaller the particle density atξ= 0 point is.The interaction of two BECs has been investigated when the trapping potential is set zero. If the trapping potential is set zero at t=0, condensates move freely along theξaxis, and the interference occurs between two BECs in the overlap region. With time increasing, the condensates'interference fringe gets stronger with the same nonlinear parameter. We also study the density with different non-linear coefficient and find that the density is maximum whenθ= 0,θ= 2π, and is zero whenθ=π.The numerical study of BEC has gone far beyond what had been introduced above. Many interesting phenomenon have been investigated by numerous methods. This thesis specialized in the numerical study of BEC by symplectic algorithm, since the symplectic algorithm has been found to be very useful in solving such problems.