# On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations

## TL;DR

This paper rigorously shows that, under long-wave and small amplitude assumptions, solutions of the improved Boussinesq equation can be approximated by two independent Camassa-Holm equations, with error estimates provided.

## Contribution

It provides a rigorous derivation and error analysis for the decoupling of the improved Boussinesq equation into two Camassa-Holm equations in the long-wave regime.

## Key findings

- Solutions split into two waves propagating in opposite directions.
- Approximation errors are quantified in terms of small parameters.
- Similar error estimates are provided for Benjamin-Bona-Mahony and Korteweg-de Vries approximations.

## Abstract

We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirectional solutions of the improved Boussinesq equation tend to associated solutions of two uncoupled Camassa-Holm equations. We give a precise estimate for approximation errors in terms of two small positive parameters measuring the effects of nonlinearity and dispersion. Our results demonstrate that, in the present regime, any solution of the improved Boussinesq equation is split into two waves propagating in opposite directions independently, each of which is governed by the Camassa-Holm equation. We observe that the approximation error for the decoupled problem considered in the present study is greater than the approximation error for the unidirectional problem characterized by a single Camassa-Holm equation. We also consider lower order approximations and we state similar error estimates for both the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.