The Cauchy problem for quasi-linear Schrödinger equations

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Mathematical Models and Numerical Methods for Bose-Einstein Condensation (2012, background): “The Cauchy problem of the Schrödinger equation with derivative nonlinearity has been investigated extensively in the literatures [115,125].”
Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, background): “These spaces were introduced earlier in the study of Schrödinger maps by two of the authors in [13], and replace the corresponding spaces in [19]–[21] (where everything was localized to finite cubes).”
Stability and instability results for standing waves of quasi-linear Schrödinger equations (2010, background): “done to develop general theories for the Cauchy problem (see nevertheless [10, 20 , 34]).”
Existence of ground states for a modified nonlinear Schrödinger equation (2009, background): “See also [18] for a result concerning very general quasilinear Schrödinger equations.”
Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data (2009, background): “Recently, the local well posedness results have been generalized to the quasi-linear (ultrahyperbolic) Schrödinger equations, see [16,17].”