LOCAL WELL-POSEDNESS FOR QUADRATIC NONLINEAR SCHRODINGER EQUATIONS AND THE "GOOD" BOUSSINESQ EQUATION

The Cauchy problem for 1-D nonlinear Schrodinger equations with quadratic nonlinearities are considered in the spaces H(s,a) defined by parallel to f parallel to H(s,a) = parallel to(1 + vertical bar xi vertical bar(s-a) vertical bar xi vertical bar(a) (f) over cap parallel to(L2), and sharp local well-posedness and ill-posedness results are obtained in these spaces for nonlinearities including the term u (u) over bar 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 >= -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) x H(s-2) with s > -1/2, which is an improvement of the previous result given by Farah (2009).