Stability of propagation features under time-asymptotic approximations for a class of dispersive equations

Abstract: 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.