The KdV, the Burgers, and the Whitham limit for a spatially periodic Boussinesq model

TL;DR

This paper rigorously justifies the use of KdV, Burgers, and Whitham equations as accurate approximations for a spatially periodic Boussinesq model with significant contrast, using Bloch wave analysis and energy estimates.

Contribution

It provides the first proof that these amplitude equations accurately predict the dynamics of a dispersive PDE in a non-small contrast periodic medium.

Findings

Validated the KdV, Burgers, and Whitham approximations over natural time scales.

Established estimates linking true solutions and amplitude equations.

First justification of these approximations in a non-small contrast periodic setting.

Abstract

We are interested in the Korteweg-de Vries (KdV), the Burgers, and the Whitham limit for a spatially periodic Boussinesq model with non-small contrast. We prove estimates between the KdV, the Burgers, and the Whitham approximation and true solutions of the original system which guarantee that these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over the natural time scales of the amplitude equations. The proof is based on Bloch wave analysis and energy estimates. The result is the first justification result of the KdV, the Burgers, and the Whitham approximation for a dispersive PDE posed in a spatially periodic medium of non-small contrast.