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

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.

💡 Research Summary

The paper addresses the initial‑value problem for a pair of coupled regularised Boussinesq (cRB) equations posed on the infinite line with rapidly decaying (localized) initial data. The system, written in terms of the primary fields (u(x,t)) and (w(x,t)), contains quadratic nonlinearity, fourth‑order dispersion (the regularising terms (u_{ttxx}) and (w_{ttxx})), and linear coupling through parameters (\delta) and (\gamma). After differentiating the original equations with respect to (x) the authors work with the gradient variables (f=u_x) and (g=w_x), which leads to a more convenient form (2) for asymptotic analysis.

The core of the analysis is a multiple‑scale expansion that separates a fast characteristic time (the linear wave propagation at speed 1 for the first field and speed (c) for the second) from a slow time (T=\varepsilon t) that captures weak nonlinearity and dispersion. Two distinct asymptotic regimes are considered:

1. Weakly distinct characteristic speeds ((c-1 = O(\varepsilon))). In this case the right‑ and left‑going components of both fields travel at almost the same speed. The authors expand the solution as \