Local well-posedness for the Zakharov system on the multidimensional torus

The initial value problem of the Zakharov system on the two dimensional torus with general period is shown to be locally well posed in the Sobolev spaces of optimal regularity, including the energy space. Unlike the one dimensional case studied by Takaoka [18], the optimal regularity does not depend on the period of torus. The proof relies on a standard iteration argument using the Bourgain norms. The same strategy is also applicable to three and higher dimensional cases.