Modulation instability and rogue waves for the sixth-order nonlinear Schro¨dinger equation with variable coefficients on a periodic background

Abstract In this paper, rogue wave solutions of a sixth-order focusing nonlinear Schr¨odinger (NLS) equation with variable coefficients are investigated on a periodic background. To get the results, we take advantage of Darboux transformation approach and the nonlinearization of spectral problem and we firstly find one kind of rogue wave solution evolves periodically with time on a periodically spatial background. Modulation instability (MI) of the sixth-order focusing NLS equation with variable coefficients is also studied.