  In this paper, we explore the anomalous dispersive relations, inverse
scattering transform and fractional N-soliton solutions of the integrable
fractional higher-order nonlinear Schrodinger (fHONLS) equations, containing
the fractional Hirota (fHirota), fractional complex mKdV (fcmKdV), and
fractional Lakshmanan-Porsezian-Daniel (fLPD) equations, etc. The inverse
scattering problem can be solved exactly by means of the matrix Riemann-Hilbert
problem with simple poles. As a consequence, an explicit formula is found for
the fractional N-soliton solutions of the fHONLS equations in the
reflectionless case. In particular, we analyze the fractional one-, two- and
three-soliton solutions with anomalous dispersions of fHirota and fcmKdV
equations. The wave, group, and phase velocities of these envelope fractional
1-soliton solutions are related to the power laws of their amplitudes. These
obtained fractional N-soliton solutions may be useful to explain the related
super-dispersion transports of nonlinear waves in fractional nonlinear media.
