  We consider solutions in frequency bands of dispersive equations on the line
defined by Fourier multipliers, these solutions being considered as wave
packets. In this paper, a refinement of an existing method permitting to expand
time-asymptotically the solution formulas is proposed, leading to a first term
inheriting the mean position of the true solution together with a constant
variance error. In particular, this first term is supported in a space-time
cone whose origin position depends explicitly on the initial state, implying
especially a shifted time-decay rate. This method, which takes into account
both spatial and frequency information of the initial state, makes then stable
some propagation features and permits a better description of the motion and
the dispersion of the solutions of interest. The results are achieved firstly
by making apparent the cone origin in the solution formula, secondly by
applying precisely an adapted version of the stationary phase method with a new
error bound, and finally by minimizing the error bound with respect to the cone
origin.
