We investigate the well-posedness theory of the 2-D fractional nonlinear
Schr\"odinger equation (NLSE) with a mixed degree of derivatives. Motivated by
models in optics and photonics where the light propagation is governed by
non-quadratic, fractional, and anisotropic dispersion profile, this paper
presents first results in this direction. Dispersive estimates are developed in
the context of anisotropic Sobolev spaces defined by inhomogeneous symbols. The
main model is shown to exhibit scattering for small data in the
scaling-critical space. Furthermore the continuity of solution with respect to
the dispersion parameter is shown on a compact time interval.Comment: Revised Articl