New paper

http://www.worldscientific.com/doi/abs/10.1142/S0219199714500345?af=R

I am not an expert, but I found this document on the internet www.math.brown.edu/~holmer/lectures/kenigi.pdf

Hope it can help you.

This is a good question.

The general quasi-linear Schroedinger equation  is considered in

X. Liu, Analytical and numerical results for some classes of nonlinear Scrhoedinger equations, Ph.D. thesis, University of Toronto, 2013:

http://www.math.toronto.edu/liuxiao/index_files/xiao_thesis.pdf

See Theorem 1.2.1, page 6, which gives a condition for the existence of a unique solution for a non-linear Schroedinger equation.

New paper

http://www.worldscientific.com/doi/abs/10.1142/S0219199714500345?af=R