Nonlinear Phenomena In One-dimensional Lattices And Bose-Einstein Condensates

Nonlinearity is a very fascinating phenomenon in the nature. From earlier fields such as fluids and plasmas to previous systems like solids, biological chemistry, geologies, universes and even society, one often observes large-amplitude wave motions, which is an important typical nonlinear phenomenon. Localized large-amplitude waves called solitons represent one of the most striking aspects of nonlinear phenomena. Soliton, like particles, can travel over long-distances without attenuation and changes of wave shapes, which is the result from the balance of the interplay between dispersion and nonlinearity. The great stability of solitons has led to numerous attempts in recent years to explain many novel phenomena in terms of solitons in different branches of physics and other sciences.In the PhD dissertation, we study nonlinear phenomena of one-dimensional lattices and Bose-Einstein condensates (BEC) by using multiple-scale method combined with quasi-discreteness approximation. As is well known, nearest-neighbor harmonic and cubic, quartic anharmonic interactions may cause nonlinear lattices exhibit series of elementary excitation, which exist not only envelope solitons but also asymmetric intrinsic localized modes: envelope-kink, and envelope-antikink solitons. This stimulates that it becomes more and more interesting and important to study the issue of lattice dynamics with higher anharmonic and long-range interactions. We study firstly soliton properties of nonlinear lattices with power-law and Kac-Baker long-range harmonic interactions, respectively. On the other hand, due to the observation of Bose-Einstein condensation in magnetically trapped atomic vapors of rubidium, sodium, and lithium in 1995, many phenomena of BEC such as nonlinear effects arising from inter-particle interactions, collective excitations, interactions between adjacent condensates, have attracted considerable attentions of scientists. In the mean field approximation, the dynamics behaviors of atoms of BEC can be described by the...