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The first target of this article is the local well-posedness question for 1Dquasilinear Schr\"odinger equations with cubic nonlinearities. The study ofthis class of problems, in all dimensions, was initiated in pioneering work ofKenig-Ponce-Vega for localized initial data, and then continued byMarzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective hereis to fully redevelop the study of this problem in the 1D case, and to prove a\emph{sharp local well-posedness} result.  The second goal of this article is to consider the long time/global existenceof solutions for the same problem. This is motivated by a broad conjectureformulated by the authors in earlier work, which reads as follows:``\emph{Cubic defocusing dispersive one dimensional flows with small initialdata have global dispersive solutions}''; the conjecture was initially provedfor a well chosen semilinear model of Schr\"odinger type.  Our work here establishes the above conjecture for 1D quasilinearSchr\"{o}dinger flows. Precisely, we show that if the problem has \emph{phaserotation symmetry} and is \emph{conservative and defocusing}, then small datain Sobolev spaces yields global, scattering solutions. This is the first resultof this type for 1D quasilinear dispersive flows. Furthermore, we prove it atthe minimal Sobolev regularity in our local well-posedness result.  The defocusing condition is essential in our global result. Without it, theauthors have conjectured that \emph{small, $\epsilon$ size data yields longtime solutions on the $\epsilon^{-8}$ time-scale}. A third goal of this paperis to also prove this second conjecture for 1D quasilinear Schr\"{o}dingerflows.

Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schr\"odinger flow