The derivation of the conservation law for defocusing nonlinear Schroedinger equations with non-vanishing initial data at infinity

For nonlinear Schr"odinger equations in less than or equal to four dimension, with non-vanishing initial data at infinity, a new approach to derive the conservation law is obtained. Since this approach does not contain approximating procedure, the argument is simplified and some of technical assumption of the nonlinearity to derive the conservation law and time global solutions, is removed.