  This paper is concerned with time global behavior of solutions to nonlinear
Schr\"odinger equation with a non-vanishing condition at the spatial infinity.
Under a non-vanishing condition, it would be expected that the behavior is
determined by the shape of the nonlinear term around the non-vanishing state.
To observe this phenomenon, we introduce a generalized version of the
Gross-Pitaevskii equation, which is a typical equation involving a
non-vanishing condition, by modifying the shape of nonlinearity around the
non-vanishing state. It turns out that, if the nonlinearity decays fast as a
solution approaches to the non-vanishing state, then the equation admits a
global solution which scatters to the non-vanishing element for both time
directions.
