Stability of solitary waves for nonlinear dispersive equations

对于一个包含激光频率起伏的光学约束下的玻色-爱因斯坦凝聚模型，从数学上证明了构造线性随机薛定谔方程基本解的解的存在性，并证明了受时间白噪声影响的稳定驻波的长时间行为。另一方面，对描述激光在空芯光纤中传输的模型的激发态稳定性进行了研究。对于光纤模型，当介质中存在两个对称缺陷时，我们发现了基态的对称性破缺。

Concerning a model for the Bose-Einstein Condensation trapped under an optical confinement involving fluctuations by laser frequencies, I justified mathematically the existence of solution constructing the fundamental solution of linear stochastic Schrodinger equations, and proved a long time behavior of stable standing waves influenced by temporal white noise. On the other hand, I investigated the stability of excited states for a model which describes the propagation of laser beam in the hollow-core fibers. As for a model of optic fibers, I showed the symmetry breaking of ground state for the case where two symmetric defects are present in the medium.