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

TL;DR

This paper demonstrates that generalized Burgers and KdV equations with quadratic nonlinearity can be formulated as geodesic equations on specific diffeomorphism groups, establishing well-posedness results for these extensions.

Contribution

It shows that these nonlinear PDEs can be interpreted geometrically as geodesic flows on diffeomorphism groups, extending the understanding of their structure.

Findings

Recasting equations as geodesic flows on diffeomorphism groups

Proving well-posedness of the Burgers equation with quadratic term

Connecting PDEs to geometric group theory

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.