Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$
Nobu Kishimoto
Communications on Pure & Applied Analysis 2008, 7(5): 1123-1143 doi: 10.3934/cpaa.2008.7.1123
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 $\bar u^2$.

keywords: Quadratic Schrödinger equation weighted norm. bilinear estimate well-posedness
Resonant decomposition and the $I$-method for the two-dimensional Zakharov system
Nobu Kishimoto
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 4095-4122 doi: 10.3934/dcds.2013.33.4095
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).
keywords: resonant decomposition. $I$-method Zakharov system global well-posedness

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