Local well-posedness for quadratic nonlinear Schrödinger equations and the ``good'' Boussinesq equation

Abstract

The Cauchy problem for 1-D nonlinear Schrödinger equations with quadratic nonlinearities are considered in the spaces $H^{s,a}$ defined by $ \| f \|_{H^{s,a}}=\| (1+|\xi|)^{s-a} |\xi|^a \widehat{f} \|_{L^2}, $ and sharp local well-posedness and ill-posedness results are obtained in these spaces for nonlinearities including the term $u\bar{u}$. In particular, when $a=0$ the previous well-posedness result in $H^s$, $s>-1/4$, given by Kenig, Ponce and Vega (1996), is improved to $s\ge -1/4$. This also extends the result in $H^{s,a}$ by Otani (2004). The proof is based on an iteration argument similar to that of Kenig, Ponce and Vega, with a modification of the spaces of the Fourier restriction norm. Our result is also applied to the ``good'' Boussinesq equation and yields local well-posedness in $H^s\times H^{s-2}$ with $s>-1/2$, which is an improvement of the previous result given by Farah (2009).