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CPAA

We prove the local well-posedness of a 1-D quadratic nonlinear Schrödinger equation

$ iu_t+u_{x x}=\bar u^2$

in $H^s(\mathbb R)$ for $s\ge -1$ and ill-posedness below $H^{-1}$. The same result for another quadratic nonlinearity $u^2$ was given by I. Bejenaru and T. Tao, *Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation*, J. Funct. Anal. **233 ** (2006), but the function space of solutions depended heavily on the special property of the nonlinearity $u^2$. We construct the solution space suitable for the nonlinearity $\bar u^2$.

DCDS

The initial value problem of the Zakharov system on a two-dimensional torus with general period is considered in this paper.
We apply the $I$-method with some `resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class.
We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).

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