Stiff microscale forcing and solitary wave refocusing
Juan Carlos Muñoz Grajales | Nachbin, Andre
solitary waves | inhomogeneous media | refocusing
This study is focused on wave propagation in heterogeneous media and on capturing the cumulative effect due to small-scale orographic features. More specifically on designing a numerical method capable of capturing these effects over long propagation distances (the macroscale) when a weakly dispersive, weakly nonlinear solitary wave (the mesoscale) is forced by a disordered microscale. An accurate and stable numerical scheme is presented for solving a variable coefficient nonlinear Boussinesq system, in the presence of highly oscillatory solutions, together with a new result on its linear stability analysis. The stability analysis focuses on describing the stiffness promoted by the topography's microscale and how it can be removed through an appropriate change of coordinates. The new mathematical formulation acts as a preconditioning at the differential equation level and the spectrum's range becomes insensitive to the microscale. The other goal of this paper is to present the time-reversal and refocusing for solitary waves. In previous works the effects of dispersion and of nonlinearity have been analysed separately. The time-reversed refocusing of solitary waves is observed numerically over a set of different simulations. Presently there is no mathematical theory for the time-reversed refocusing of solitons. Hopefully these simulations will shed some light in this direction.