  In this paper we consider the Schr{\"o}dinger equation with nonlinear
derivative term. Our goal is to initiate the study of this equation with non
vanishing boundary conditions. We obtain the local well posedness for the
Cauchy problem on Zhidkov spaces X k (R) and in $\phi$ + H k (R). Moreover, we
prove the existence of conservation laws by using localizing functions.
Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.
