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

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.