A Study of a (3+1)-Dimensional Hirota Bilinear Model with Variable Coefficients: Generalized Higher-Order Rogue Waves and Wronskian Solutions

The Hirota bilinear equation with variable coefficients (VCs) serves as a fundamental model for capturing nonlinear wave dynamics in fluids and oceans. Utilizing the Hirota bilinear framework alongside advanced symbolic computation techniques, higher-order rational solutions to this equation have been comprehensively investigated. These solutions unveil a wide range of waveforms, including multi-rogue waves (RWs) and multi-parallel solitons. To further expand the solution space, the rational solutions have been generalized by introducing two adjustable parameters, alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, referred to as center control parameters. These parameters enable the generation of a versatile class of nonlinear, controllable RWs, enhancing the flexibility and applicability of the model. Soliton solutions have also been constructed using the Wronskian method, with their validity rigorously established through proofs grounded in Pl & uuml;cker relations. By employing various functional forms for the VCs, the dynamic properties of these solutions have been extensively analyzed and visualized through three-dimensional representations, showcasing their rich complexity and diverse behaviors.