A complete description of properties of the wave-train bifurcating from unstable basic
oscillatory states (CW nonlinear stationary states) of the nonlinear Schrödinger equation
are studied in the moving frames of reference as an initial value problem and using the
methods of absolute and convective instabilities. The predictions are in excellent
agreement with numerical solutions and may contribute understanding the nonlinear
Schrödinger equation complex dynamics under various initial conditions including,
localized and/or noisy initial conditions