Breathers and rogue waves for an eighth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber

An eighth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber is investigated in this paper. Through the Darboux transformation (DT) and the generalized DT, we obtain the breather and rogue wave solutions. Choosing different values of α(x), β(x), ; (x), δ(x),ε(x), ζ(x), and η(x),, which are the coefficients of the group velocity dispersion, the third-, fourth-, fifth-, sixth-, seventh-, and eighth-order dispersions, respectively, we investigate their effects on the solutions, where x is the scaled propagation variable. Interaction between two kinds of the breathers is studied, i.e., the Akhmediev and Kuznetsov-Ma breathers, and we find that the interaction regions are similar to those of the second-order rogue waves. When α(x), β(x), ; (x), δ(x),ε(x), ζ(x), and η(x), are all chosen as x, x², x³, respectively, peaks of the rogue waves are found to have the parabolic, cubic, and quasi-parabolic shapes. The Akhmediev breathers are seen to pass through the Kuznetsov-Ma breathers who change their phases after the interaction. Rogue waves can be split into some first-order rogue waves or complex structures when the periodic and odd functions are chosen for ε(x), ζ(x), and η(x).