Complex nonlinearities of rogue waves in generalized inhomogeneous higher-order nonlinear Schrödinger equation

In this paper, the Nth-order rogue waves are investigated for an inhomogeneous higher-order nonlinear Schrödinger equation. Based on the Heisenberg ferromagnetic spin system, the higher-order nonlinear Schrödinger equation is generated. The generalized Darboux transformation is constructed by the Darboux matrix. The solutions of the Nth-order rogue waves are given in terms of a recursive formula. There are complex nonlinear phenomena in the rogue waves, add the first-order to the fourth-order rogue waves are discussed in some figures obtained by analytical solutions. It is shown that the general Nth-order rogue waves contain 2n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2n-1$$\end{document} free parameters. The free parameters play a crucial role to affect the dynamic distributions of the rogue waves. The results obtained in this paper will be useful to understand the generation mechanism of the rogue wave.