  The purpose of this article is to clarify the Cauchy theory of the water
waves equations as well in terms of regularity indexes for the initial
conditions as for the smoothness of the bottom of the domain (namely no
regularity assumption is assumed on the bottom). Our main result is that,
following the approach developped by T. Alazard and G. M\'etivier in [1], after
suitable paralinearizations, the system can be arranged into an explicit
symmetric system of Schr\"odinger type. We then show that the smoothing effect
for the (one dimensional) surface tension water waves proved by H.
Christianson, V. M. Hur, and G. Staffilani in [9], is in fact a rather direct
consequence of this reduction, which allows also to lower the regularity
indexes of the initial data, and to obtain the natural weights in the
estimates.
