On Coupled Boussinesq Equations and Ostrovsky equations free from zero-mass contradiction

Abstract

Long weakly-nonlinear waves in a layered waveguide with an imperfect interface (soft bonding between the layers) can be modelled using coupled Boussinesq equations. Previously, we considered the case when the materials of the layers have close mechanical properties, and the system supports radiating solitary waves. Here we are concerned with a more challenging case, when the mechanical properties of the materials of the layers are significantly different, and the system supports wave packet solutions. We construct a weakly-nonlinear solution of the Cauchy problem for this system, considering the problem in the class of periodic functions on an interval of finite length. The solution is constructed for the deviation from the evolving mean value using a novel multiple-scales procedure involving fast characteristic variables and two slow time variables. By construction, the Ostrovsky equations emerging within the scope of this procedure are solved for initial conditions with zero mean while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We also discuss the application of the solution to the study of co-propagating waves generated by the solitary or cnoidal wave initial conditions, as well as the case of counter-propagating waves and the resulting wave interactions. One local and two nonlocal conservation laws ae obtained and used to control the accuracy of the numerical simulations.