Validity of the weakly-nonlinear solution of the Cauchy problem for the Boussinesq-Ostrovsky equation

TL;DR

This paper analyzes the validity of a weakly-nonlinear asymptotic solution for the periodic Boussinesq-Ostrovsky equation, demonstrating improved accuracy and error control in energy space through analytical and numerical methods.

Contribution

It provides a rigorous justification of the weakly-nonlinear solution incorporating linearized Ostrovsky solutions, with enhanced error estimates in the energy space.

Findings

Improved asymptotic accuracy at Ostrovsky time scales.

Error control in the energy space for periodic functions.

Inclusion of nonzero mean values in the analysis.

Abstract

We consider the initial-value problem for the regularized Boussinesq-Ostrovsky equation in the class of periodic functions. Validity of the weakly-nonlinear solution, given in terms of two counter-propagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the time scales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis.