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International Journal of Mathematics and Mathematical Sciences
Volume 9, Issue 2, Pages 387-396

Conservation laws for shallow water waves on a sloping beach

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

Received 20 February 1985

Copyright © 1986 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.


Shallow water waves are governed by a pair of non-linear partial differential equations. We transfer the associated homogeneous and non-homogeneous systems, (corresponding to constant and sloping depth, respectively), to the hodograph plane where we find all the non-simple wave solutions and construct infinitely many polynomial conservation laws. We also establish correspondence between conservation laws and hodograph solutions as well as Bäcklund transformations by using the linear nature of the problems on the hodogrpah plane.