# American Institute of Mathematical Sciences

May  2015, 35(5): 2123-2130. doi: 10.3934/dcds.2015.35.2123

## One-parameter solutions of the Euler-Arnold equation on the contactomorphism group

 1 Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, United States

Received  May 2014 Revised  August 2014 Published  December 2014

We study solutions of the equation $g_t-g_{tyy} + 4g^2 - 4gg_{yy} = y gg_{yyy}-yg_yg_{yy}, \qquad y\in\mathbb{R},$ which arises by considering solutions of the Euler-Arnold equation on a contactomorphism group when the stream function is of the form $f(t,x,y,z) = zg(t,y)$. The equation is analogous to both the Camassa-Holm equation and the Proudman-Johnson equation. We write the equation as an ODE in a Banach space to establish local existence, and we describe conditions leading to global existence and conditions leading to blowup in finite time.
Citation: Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123
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