As we well known that, the Darboux transformation is a powerful method to generate some interesting exact solutions from some trivial solutions for the integrable equations. For instance, through the zero solution of vNLSE, the multi-soliton solutions can be constructed. In recent years, there are lots of works which focus on generating the exact solutions by the plane wave seed solution (genus zero solution). Some physical solutions---the Akhmediev breather, K-M breather, rogue wave solution, dark soliton, bright-dark soliton and so on can be readily constructed. In this talk, we give the Darboux transformations for integrable vNLSE with different types. Furthermore, we give the exact solutions for the vNLSE on the non-vanishing background by some analysis on the formulas. Here, we stress that the inverse scattering theory of vNLSE for the non-vanishing background was not solved completely under the general plane wave background. But by applying the Darboux transformation, the exact solutions and relationship with the discrete spectrum are clear, which means that the discrete spectrum part of inverse scattering theory is solved. Speaker(s): Liming Ling (South China University of Technology)