A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations

Abstract

We show that two families of equations on the real line, the generalized inviscid Proudman--Johnson equation, and the $r$-Hunter--Saxton equation (recently introduced by Cotter et al.) coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman--Johnson equations as geodesic equations of right invariant homogeneous $W^{1,r}$-Finsler metrics on an appropriate diffeomorphism group on $\mathbb{R}$. Generalizing a construction of Lenells for the Hunter--Saxton equation, we analyze the $r$-Hunter--Saxton equation using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby we show that the periodic case is equivalent to the geodesic equation on the $L^r$-sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior.

arXiv commentversion 3: corrected an error regarding the equivalence of the equations in the periodic case (there is equivalence only in the non-periodic case). Abstract and introduction were changed accordingly. version 2: minor changes

References in corpus (14)

• An integrable shallow water equation with peaked solitons
• The Camassa-Holm equation as a geodesic flow on the diffeomorphism group
• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• The Degasperis-Procesi equation as a non-metric Euler equation
• The geometry of a vorticity model equation
• A zoo of diffeomorphism groups on ${\mathbb{R}}^n$
• Geodesic Equations on Diffeomorphism Groups
• The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line
• Blow-up of Solutions to the Generalized Inviscid Proudman-Johnson Equation
• An exotic zoo of diffeomorphism groups on ${\mathbb{R}}^n$
• Half-Lie groups
• The r-Hunter-Saxton equation, smooth and singular solutions and their approximation
• A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations
• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

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