New perspective on fractional Hamiltonian amplitude equation

In this paper, we present a pioneering investigation on the fractional Hamiltonian amplitude equation involving the beta fractional derivative for the first time, addressing a research gap in the field of nonlinear fractional dynamics. Our primary objective is to develop effective analytical techniques capable of solving the fractional Hamiltonian amplitude equation and obtaining novel soliton solutions. To achieve this, we introduce two advanced methods: the extended fractional rational sineδ-cosineδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sin e_{\delta } - \cos ine_{\delta }$$\end{document} and the fractional rational sinhδ-coshδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sinh_{\delta } - \cosh_{\delta }$$\end{document} techniques. By employing these cutting-edge approaches, we successfully derive new types of soliton solutions, demonstrating the reliability and efficiency of the proposed methods. Furthermore, the applicability of these techniques extends to various fractional nonlinear evolution models, highlighting their versatility in the realm of fractional dynamics. Finally, we provide a comprehensive presentation of the results, which substantiate the effectiveness of the methods in solving the complex fractional Hamiltonian amplitude equation.