Modulated Amplitude Waves In A Multi-component Bose-einstein Condensate

Multi-component Bose-Einstein condensate is modeled by a coupled system of GP equations in the frame of mean field theory. In this paper, we study a multi-component Bose-Einstein condensate in optical lattices, and establish the existence and multiplicity of modulated amplitude waves by using averaging theory and Hamiltonian perturbation theory. The main contents of the dissertation are as follows:1、Modulated amplitude waves in binary Bose-Einstein condensateBased on a mean-field limit, when the temperature is much lower than the critical temperature, we study the system of two-component Bose-Einstein condensate Where e is a small parameter. Amplitude and phase equations of the system are derived by the uniformly propagating coherent structures with the ansatz, which are coupled by nonlinear interactions with singularities. All the solutions of the unperturbed system(e(28)0) of amplitude equations are quasi-periodic with multiple irrational frequencies except one rest point. First of all, the existence of periodic solutions for amplitude equations is proved by using some results from averaging theory. Then,combining with Hamiltonian perturbation theory, we establish the existence of infinitely many MAWs with non-trivial phases in binary Bose-Einstein condensate.2、Modulated amplitude waves in three-component Bose-Einstein condensateconsider the model of three-component Bose-Einstein condensate Where are the sinusoidal optical lattice potential. Based on the implicit function theorem and the Lyapunov-Schmidt reduction, we establish the existence of modulated amplitude waves by means of averaging theory and Hamiltonian perturbation theory in three-component Bose-Einstein condensate.