Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

TL;DR

This paper derives a hierarchy of Ostrovsky equations from coupled Boussinesq equations to model weakly nonlinear wave dynamics, providing analytical solutions and numerical validation for different regimes.

Contribution

It introduces a novel asymptotic approach linking Boussinesq and Ostrovsky equations, extending understanding of coupled wave systems and their weakly nonlinear solutions.

Findings

Derived a hierarchy of Ostrovsky equations for unidirectional waves.

Constructed weakly nonlinear solutions without secular growth.

Numerical simulations confirmed different solution behaviors in various regimes.

Abstract

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scales expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.