The Burgers equation and the Korteweg-de Vries equation with quadratic nonlinearity

2011/07/20 by Martin Kohlmann, Kohlmann, Martin
Mathematics · Physics and Astronomy · #35G25 #35Q53 #58D05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math.AP #msc:35G25 #msc:35Q53 #msc:58D05

paper · pdf · doi:10.48550/arxiv.1107.3994

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openalex publication_date 2011/07/20 · arxiv created 2012/04/11 · arxiv updated 2012/04/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study generalized variants of the Burgers equation and the KdV equation on the circle. The main goal of the paper is to show that both extensions can be recast as geodesic equations on a suitable diffeomorphism group of the circle and the corresponding Bott-Virasoro group respectively. As a consequence we obtain that the initial value problem for the Burgers equation with an additional quadratic term is well-posed on a scale of Sobolev spaces on the circle.

Citations

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations

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