Gap Solitons And Their Stabilities In Quasi One-dimensional BEC Under Periodic Potential

Bose-Einstein Condensate(BEC)is a new ultra-low temperature quantum state called the fifth state of matter.BEC has been implemented experimentally one after another,and researchers are very interested in BEC material wave soliton phenomenon.The Gross-Pitaevskii(GP)equation can be used to describe the dynamic behavior of the rare BEC at ultra-low temperature.In this paper,the gap solitons in quasi one-dimensional BEC with two-and three-body interactions are studied and some interesting results are obtained.The main contents are arranged as follows:The first chapter briefly introduces the development of BEC theory and its realization in experiment,then introduces the mean field theory and GP equation,and finally introduces the research situation of soliton experiment and theory in BEC.In the second chapter,the theoretical model mainly includes the formulation of quasi1-d 3-5 GP equation and the background knowledge.The multi-scale method was used to analyze the system theoretically.The GPE was transformed into a certain non-linear Nonlinear Schrodinger Equation(NLSE),and the expression of the band gap solitons was obtained.The analysis shows that the amplitude of the solitons follows the two-body or the three-body interaction increases and decreases.The third chapter main content is introduced Newton conjugate gradient method,a detailed discussion from existing in the system of the Band gap soliton under:(1)Considering the interaction among the three body out of the Band edge of the fork Band gap soliton,there are two types of fundamental soliton:on-site and off-site soliton,among them off-site can be divided into phase according to the phase difference and interphase dipole soliton,the numerical results also show that,with the increase of nonlinear interaction strength of the Band gap soliton amplitude decreases,and it accord with the results of the analysis of the multiple scales method;(2)Solitons that are not bifurcated from Band edge under two-body interaction and three-body interaction are considered respectively.(3)introduce the method of studying soliton stability:linear stability analysis and dynamic evolution.The stability of solitons is discussed in detail by using two methods.It is found that there are both stable bandgap solitons and unstable bandgap solitons in the system.Then we discusse the influence of the intensity of the two-body or three-body interaction and the external potential modulus on the stability of the soliton.It is found that there is a band-gap soliton,in which the soliton is dynamically unstable when only the two-body interaction is considered,while the soliton is stable when the three-body interaction is considered.We can see that there exists a critical value_cq for certainm.Whenq(29)q_c,the solitons are linearly stable.While the solitons are linearly unstable whenq(27)q_c.Therefore,the modulus plays an important role in the stability of the gap soliton.The dynamic behavior of gap solitons can be changed by adjusting the external potential modulus.In addition to the on-site and off-site solitons,another type of solitons in this system is called sub-basic solitons.The stability of this type of solitons is further studied.Finally,the work of this paper is summarized and the future research work is prospected.