Two-dimensional ‘discrete hydrodynamics’ and Monge–Ampere equations

An integrable discrete-time Lagrangian system on the group of area-preserving plane diffeomorphisms SDiff (R2) is considered. It is shown that non-trivial dynamics exists only for special initial data and the corresponding mapping can be interpreted as a Backlund transformation for the (simple) Monge–Ampere equation. In the continuous limit, this gives the isobaric (constant pressure) solutions of the Euler equations for an ideal fluid in two dimensions. In the Appendix written by E. V. Ferapontov and A. P. Veselov, it is shown how the discrete system can be linearized using the transformation of the simple Monge–Ampere equation going back to Goursat.