Large data local well-posedness for a class of KdV-type equations II

We consider the Cauchy problem for an equation of the form \partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms and no quadratic uu_{xx} term. For a polynomial nonlinearity with no quadratic terms, Kenig-Ponce-Vega proved local well-posedness in H^s for large s. In this paper we prove local well-posedness in low regularity Sobolev spaces and extend the result to certain quadratic nonlinearities. The result is based on spaces and estimates similar to those used by Marzuola-Metcalfe-Tataru for quasilinear Schrodinger equations.