Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrödinger equation

Modulation instability, rogue waves and spectral analysis are investigated for the nonlinear Schrödinger equation with the higher-order terms. The modulation instability distribution characteristics from the sixth-order to eighth-order nonlinear Schrödinger 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 N-order rogue wave is given. Dynamic phenomena of the first- to third-order rogue waves are illustrated, which exhibit meaningful structures. Two arbitrary parameters play important roles in the rogue wave solution. One parameter can control the width and crest deflection of rogue wave, 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 structures, including fundamental, circular and triangular patterns, are displayed and discussed. Through spectral analysis on the first-order rogue wave, when these parameters satisfy certain conditions, it occurs a transition between W-shaped soliton and rogue wave.