Abstract
The classical shallow water equations express the change with time of the height
h and the velocity ν of a 1-dimensional fluid:
νξ
νt
+
νξ
νx
+
νh
νx
=0.
νh
νx
+
νhν
νx
=0
. They possess an infinite number of integrals of motion due to Benney [1973] and can be written in Hamiltonian form relative to a symplectic structure introduced by Manin [1978]. The present paper deals with their complete integrability up to the advent of shocks. This is proved in the small under an extra assumption satisfied by most height-velocity pairs: that
h
h′ = ± ν′
only at isolated points.