initial data at a low regularity level.

Recasting the Zakharov model as a first-order system by setting $n^\pm = n \pm i D^{-1} n_t$, where $D = (-\Delta)^\frac12$, we obtain $$\label{eq:ZS} 
\begin{cases}
iu_t + \Delta u = \frac12(n^+ + n^-)u, \qquad x \in{\mathbb{T}}, \; t \in {\mathbb{R}}, \\
in^\pm_t \mp Dn^\pm = \pm D|u|^2,\\
u(x,0) = u_0(x),\\
n^\pm(x,0) = n^\pm_0(x).
\end{cases}$$ The Zakharov system conserves the Schrödinger part mass and a Hamiltonian energy: $$M(u) = \| u\|_{L^2}^2 \qquad \qquad E(u,n^\pm) = \| u_x \|_{L^2}^2 + \frac12 \Bigl( \| n^+\|_{L^2}^2 +  \| n^-\|_{L^2}^2 \Bigr)   + \frac12 \int (n^+ + n^-) |u|^2 {\,\operatorname{d}\!}x.$$ Note also that the mean values of $n$ and $n_t$ are preserved under the flow. Thus we may without loss of generality assume $n_0$ and $n_1$ are mean-zero. Sharp local well-posedness on ${\mathbb{T}}$ holds for initial data $$(u_0, n^\pm_0) \in H^\frac12({\mathbb{T}}) \times L^2({\mathbb{T}}).$$ This result is due to Takaoka [@Taka], building on the result of Ginibre-Tsutsumi-Velo [@GTV] for the Euclidean space system. Well-posedness was proved used the Fourier restriction norm technique introduced in [@Bour2].

The local result in sharp in the class of $L^2$-based Sobolev spaces. An earlier work of Bourgain [@Bour1] gives well-posedness in certain Fourier-Lebesgue type spaces. 