Theoretical Study Of Low Dimensional Periodic Bose Nolinear System Dynamics

With the development of the Laser and low temperature refrigeration technology,various new artificial low-dimensional systems have been experimentally realized.Tytical examples include Bose-Einstein condensates in one dimensional optical lattices,one dimensional waveguide arrays, the quantum wells in a quasi two-dimensionalstructure etc. These low dimensional structures exhibit novel electronic and opticalproperties. Apart from periodicity, these systems are characterized by nonlinearity.Investigation of these low dimensional properties of nonlinear periodic system hasimportant significance. The dissertation studies the prosperity of three kinds of lowdimensional nolinear periodic systems. The stability of gap solitons of strong interactiveBose gas in one-dimensional optical lattices is judged. The theoretical explanation aboutlocalization-delocalization transition (LDT) of indirect excitons in lateral electrostaticlattices is also presented.First, the gap solitons and nonlinear Bloch waves of repulsively interacting bosonsin one-dimensional optical lattices are studied, taking into account the interaction fromthe weak to the strong limits. It is shown that composition relation between the gapsolitons and nonlinear Bloch waves exists for the whole span of the interaction strength.The linear stability analysis indicates that the gap solitons are stable when their energiesare near the bottom of the linear Bloch band gap. By increasing the interaction strength,the stable gap solitons can turn into unstable. It is argued that the stable gap solitons caneasily be formed in a weakly interacting system with energies near the bottoms of thelower-level linear Bloch band gaps. It is showed the nonlinear Bloch waves can beviewed as infinite chains of gap solitons.Secondly, the existence and stability of gap solitons of the strong attractive gas bythe recent experimental realization in one-dimensional optical lattice is studied. Thecomposition relations of various gap solitons and nonlinear Bloch waves are exploredtaking into account the interplay between periodic potential and nonlinear interaction.The unstable gap solitons can become stable by increasing the amplitude of periodic potential or decreasing the nonlinear interaction. A stronger attractively interaction willbe benefit to be stable of gap solitons. The reason why gap solitons in the higher-bandgap (higher-family) can easily form near the bottom of the linear Bloch band gaps isexplained. It is confirmed that the fundamental gap solitons can be viewed as buildingblocks for complex gap waves and multiple periodic waves in the strong attractive gas.Finally, pattern distribution of indirect excitons in lateral electrostatic lattices, themodel that the interaction between indirect excitons involves both attractive two-bodyand repulsive three-body interactions is raised. The effect between periodic potential andnonlinear interactions of indirect excitons on the localization-delocalization transition istheoretically investigated. The relation between the periodic potential strength andcrucial transition density is analyzed. The indirect excions in the system is highlyquantum degenerate and macroscopic wave function can effectively describe the spatialdistribution of indirect exciton. The lower energy degenerate states of cold exitonsdominate the complex pattern is confirmed, and that the smooth component of thephotoluminescent energy is mainly decided by the periodic potential and interactionenergy.