The Korteweg–de Vries, Burgers and Whitham limits for a spatially periodic Boussinesq model. Issue 1 (30th August 2018)

• Record Type: Journal Article
• Authors: Bauer, Roman; Düll, Wolf-Patrick; Schneider, Guido
• Is Part Of: Proceedings. Volume 149:Issue 1(2019)
• Page Start: 191
• Page End: 217
• Publication Date: 2018-08-30
• Subjects: spatially periodic Boussinesq model, -- Korteweg–de Vries (KdV) equation, -- Burgers equation, -- Whitham system, -- approximation; 35Q53; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210518000227
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : We are interested in the Korteweg–de Vries (KdV), Burgers and Whitham limits for a spatially periodic Boussinesq model with non-small contrast. We prove estimates of the relations between the KdV, Burgers and Whitham approximations and the true solutions of the original system that guarantee these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over their natural timescales. The proof is based on Bloch wave analysis and energy estimates and is the first justification result of the KdV, Burgers and Whitham approximations for a dispersive partial differential equation posed in a spatially periodic medium of non-small contrast.