Weakly-Nonlinear Solution of Coupled Boussinesq Equations and Radiating Solitary Waves

Abstract

Weakly-nonlinear waves in a layered waveguide with an imperfect interface (soft bonding between the layers) can be modelled using coupled Boussinesq equations. We assume that the materials of the layers have close mechanical properties, in which case the system can support radiating solitary waves. We construct a weakly-nonlinear d'Alembert-type solution of this system, considering the problem in the class of periodic functions on an interval of finite length. The solution is constructed using a novel multiple-scales procedure involving fast characteristic variables and two slow time variables. Asymptotic validity of the solution is carefully examined numerically. We also discuss the limiting case of an infinite interval for localised initial conditions. The solution is applied to study interactions of radiating solitary waves.

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

“…The first case when c − 1 = O (ε) was considered in [25], so we only summarise the results here. In this case the waves are resonant and an initial solitary wave solution in both layers will evolve into a radiating solitary wave; a solitary wave with a co-propagating one-sided oscillatory tail [10].…”

“…We now confirm the validity of the constructed expansions by numerically solving the system (1.1) -(1.2) and comparing this direct numerical solution to the constructed solution (2.17) and (2.18) with an increasing number of terms included. The first case, when c − 1 = O (ε), was analysed in [25]. Here we determine the validity of the expansion in the second case, when c − 1 = O (1).…”

“…In order to numerically solve these equations, in this section and the subsequent section, we implement three pseudospectral numerical schemes. For the coupled Boussinesq equations we use the technique outlined in [25,26], while for the case of a single Ostrovsky equation we use the method in [21]. In [25] a pseudospectral method was outlined for coupled Ostrovsky equations, however we present a modified scheme here, based upon [27], to allow for larger time steps to be taken.…”

“…For the coupled Boussinesq equations we use the technique outlined in [25,26], while for the case of a single Ostrovsky equation we use the method in [21]. In [25] a pseudospectral method was outlined for coupled Ostrovsky equations, however we present a modified scheme here, based upon [27], to allow for larger time steps to be taken. This is presented in Appendix A.…”

“…The first case, when c − 1 = O(ε), was constructed in [25] and is briefly overviewed at the beginning of Section 2. We will re-examine this solution in comparison with the more complicated case when c − 1 = O(1), which is the main topic of the present paper.…”