| Bose-Einstein condensation (BEC) is one of the most important results in quantum statistics. It is also the theoretical basis of many macroscopic quantum phenomena, such as superconductivity and superfluidity. After 70 years of Einstein's theoretical prediction based on Bose's work, the BEC of weakly interacting Bose gases (dilute alkali gases) was demonstrated at last experimentally in 1995. The realization of BEC is of great significance not only for basic theoretical research, but also of very promising applications in technology, such as the realization of an atom laser. In this thesis we study the interaction of the elementary excitations generated in BECsThe first part of the thesis is the investigation of three-wave resonant interaction of the elementary excitations created in a disk-shaped BEC. If in the condensate the atoms have a stronger confinement in the z-direction, we can reduce a three-dimensional problem to a two-dimensional one. We show that the phase-matching conditions can be fulfilled by suitably choosing the wave vectors and frequencies and hence the three-wave resonance of the excitations in condensate background is possible to realize. Starting from the Gross-Pitaevskii equation, which is valid to describe weakly interacting Bose gases at zero-temperature, we derive the nonlinearly-coupled envelope equations controlling the three-wave resonant interaction. We provide three-wave soliton solutions and check their stability using numerical simulations.The second part of the thesis studies the four-wave mixing of BEC matter waves. Under phase-matching conditions and using a method of multiple-scales, we derive the nonlinear envelope equations for the four-wave mixing. We give the exact solutions expressed by Jacob elliptic functions. Our results agree quite well with theexperimental ones.The results obtained in this thesis are useful not only for the understanding of the dynamic characters of the excitations, but also for practical applications of BECs.
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