Cauchy theory for the water waves system in an analytic framework

In this paper we consider the Cauchy problem for gravity water waves, in a domain with a flat bottom and in arbitrary space dimension. We prove that if the data are of size $\epsilon$ in a space of analytic functions which have a holomorphic extension in a strip of size $\sigma$, then the solution exists up to a time of size $C\mathrm{/}\epsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma -C^{\mathrm{\prime }}\epsilon t$.