Abstract
In this article, we consider the Cauchy problem associated with the Zakharov system on the torus. We obtain unconditional uniqueness of solutions in low regularity Sobolev spaces including the energy space in one and two dimensions. We also prove convergence of solutions in the energy space, as the ion sound speed tends to infinity, to the solution of a cubic nonlinear Schrodinger equation, for dimensions one and two. Our proof of unconditional uniqueness is based on the method of infinite iteration of the normal form reduction; actually, we simply show a certain set of multilinear estimates, which was proposed as a criterion for unconditional uniqueness in [13]. The convergence result is obtained by a similar argument to the non-periodic case [13], which uses conservation laws and unconditional uniqueness for the limit equation.