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

Florent Dewez (1)

1 MODAL - MOdel for Data Analysis and Learning

We consider solutions of dispersive equations on the line defined by Fourier multipliers with initial data having compactly supported Fourier transforms. In this paper, a refinement of an existing method permitting to expand time-asymptotically the solution formulas is proposed. Here the first term of the expansion is supported in a space-time cone whose origin depends explicitly on the initial datum. As an important consequence of our refined method, the first term inherits the mean position of the solution together with a constant variance error and a shifted time-decay rate is obtained. Hence this refinement, which takes into account both spatial and frequency information of the initial datum, makes stable some propagation features under time-asymptotic approximations and permits a better description of the time-asymptotic behaviour of the solutions. 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 minimising the error bound with respect to the cone origin.

Mots clés: Wave packet; Dispersive equation; Oscillatory integral; Stationary phase method; Frequency band

Domaines: Mathématiques [math]

Florent Dewez. Stability of propagation features under time-asymptotic approximations for a class of dispersive equations. Journal of Mathematical Analysis and Applications, 2020, 491 (1), pp.124292. ⟨10.1016/j.jmaa.2020.124292⟩. ⟨hal-03490225⟩