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International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 3, Pages 557-562
http://dx.doi.org/10.1155/S016117128700067X

The shallow water equations: conservation laws and symplectic geometry

Department of Mathematical Sciences, University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 20 February 1985

Copyright © 1987 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the system of nonlinear differential equations governing shallow water waves over a uniform or sloping bottom. By using the hodograph method we construct solutions, conservation laws, and Böcklund transformations for these equations. We show that these constructions are canonical relative to a symplectic form introduced by Manin.