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          | In this talk, we present some recent results on the stability of traveling waves of nonlinear Schr\"{o}dinger equation with nonzero condition at infinity. Two important physical models are Gross-Pitaevskii  equation and NLS with cubic-quintic nonlinearity. These results are obtained in collaboration with Zhiwu Lin  and  Chongchun Zeng. | 
         
     
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