September  2008, 7(5): 1123-1143. doi: 10.3934/cpaa.2008.7.1123

Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$

1. 

Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Received  October 2007 Revised  April 2008 Published  June 2008

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 $\bar u^2$.

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