  Modulation instability, rogue wave and spectral analysis are investigated for
the nonlinear Schrodinger equation with the higher-order terms. The modulation
instability distribution characteristics from the sixth-order to the
eighth-order nonlinear Schrodinger equations are studied. Higher-order
dispersion terms are closely related to the distribution of modulation
stability regime, and n-order dispersion term corresponds to $n-2$ modulation
stability curves in the modulation instability band. Based on the generalized
Darboux transformation method, the higher-order rational solutions are
constructed. Then the compact algebraic expression of the Nth-order rogue wave
is given. Dynamic phenomena of first- to third-order rogue waves are
illustrated, which exhibit meaningful structures. Two arbitrary parameters play
important roles in the rogue wave solution. One can control deflection of crest
of rogue wave and its width, while the other can cause the change of width and
amplitude of rogue wave. When it comes to the third-order rogue wave, three
typical nonlinear wave constructions, namely fundamental, circular and
triangular are displayed and discussed. Through the spectral analysis on
first-order rogue wave, when these parameters satisfy certain conditions, it
occurs a transition between W-shaped soliton and rogue wave.
