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

Nobu Kishimoto

doi:10.3934/cpaa.2008.7.1123

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2008, Volume 7, Issue 5: 1123-1143. Doi: 10.3934/cpaa.2008.7.1123

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

Nobu Kishimoto^1,

1.
Department of Mathematics, Kyoto University, Kyoto 606-8502

Received: September 30, 2007

Revised: March 31, 2008

Published: August 2008

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Abstract

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 $\bar u^2$.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35G25, 46E35.

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

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