Under a non resonance condition, we establish the unique local existence results in weak function spaces for the initial value problem of the Zakharov system with the periodic boundary condition. The proof is based on the Fourier restriction norm method, which was developed by J. Bourgain and C.E. Kenig, G. Ponce and L. Vega.
"Well-posedness for the Zakharov system with the periodic boundary condition." Differential Integral Equations 12 (6) 789 - 810, 1999. https://doi.org/10.57262/die/1367241476