| Bose-Einstein Condensate.a brand new state of matter.has become a very interesting frontier of physics in recent years. Due to the observations of dark and bright matter wave solitons and their potential applications in the Bose-Einstein Condensates, the study of matter wave solitons has been one of the host issues in the fields of both low-temperature physics and condensate matter physics. Among the various models, a system of the hard-core bosons (HCB) gas has been used to describe the limit of a strongly repulsive interaction Bose-Einstein Condensate system. The hard-core bosons system belongs to the strongly corre-lated quantum systems and the Hard-Core Bose-Hubbard model is one of such basic models. Until now. many people have studied this strongly repulsive inter-action Bose-Einstein Condensate system and obtained the bright, dark and gap solitons with the numeric method. However, few analytically exact solutions of this system.especially the case that not being restricted to one dimension, have been investigated.The classical Lie-group method is effective for searching analytically the exact solutions of non-integrable system. In this paper, using the classical Lie-group method, we separately study one and two dimensional equations of motion of the strongly repulsive Bose-Einstein Condensate system known as the hard-core bosons with nearest-neighbor interactions which may mimic certain aspects of the long range dipole-dipolc interactions. We obtain sets of analytically exact solutions including dark soliton, bright soliton and periodic solutions for the mat-ter waves in the one and two dimensional condensates separately. Furthermore, we investigate the properties of these matter waves propagation, elucidating that adjusting the optical lattice potential strength how to effect the solutions char-acteristics, including the forms, the amplitudes and the propagation velocities of these matter waves separately. The main contents of this paper are as follows:First, introducing the related background knowledge of the Bose-Einstein Condensate involving the concept, development process, the basic theories and the models. We also introduce the development process of matter wave solitons research in the Bose-Einstein Condensates. In this paper, we begin the work with the extended lattice Bose-Hubbard model and get equations of motion for the strongly repulsive Bose-Einstein Condensate by using the spin-coherent state averages, as differs from many earlier studies considerably.Secondly, using the classical Lie-group method.we study two cases that equations in the one and two dimension separately, obtaining the corresponding exact solutions including soliton and periodic solutions for the matter waves in the strongly repulsive Bose-Einstein Condensates. We analytically study the matter waves dynamics behaviors of condensates trapped in the optical lattice potential, elucidating that modulating the optical lattice potential strength how to effect the matter waves in the condensates.Finally, in the forth part of the paper, we present a conclusion for our work and give some outlooks for the further works in the field of the strongly repulsive Bose-Einstein Condensate system. |
