Analytical Soliton Solutions to a (2+1)-Dimensional Variable Coefficients Graphene Sheets Equation Using the Application of Lie Symmetry Approach: Bifurcation Theory, Sensitivity Analysis and Chaotic Behavior

The prime objective of this paper is to investigate the variable coefficients (2+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+$$\end{document}1)-dimensional graphene sheets (vcGS) equation, which characterizes the thermophoric wave motion of wrinkles in graphene sheets. Wide-ranging applications of this equation are found across various domains, including 2D optics, nanophotonics, and nanoelectronics. Two distinct methods, namely the Lie symmetry approach and generalized exponential differential rational function (GEDRF), are employed to explore new solitary waves within the considered equation. The initial approach involves transforming the vcGS equation into a set of nonlinear ordinary differential equations using the Lie group transformation technique. Subsequently, the resulting ordinary differential equations are analytically solved to get the invariant solutions of the governing equation. The second approach utilizes a generalized exponential differential rational function (GEDRF) method to convert the variable coefficients graphene sheets equation directly into an ordinary differential equation. A range of solutions, including hyperbolic functions, trigonometric functions, and more general solutions expressed in terms of arbitrary functions and constants, is obtained. By appropriately assigning values to free parameters, the dynamic wave structures of certain analytical solutions are illustrated through two-dimensional, three-dimensional, and contour graphics. A variety of phenomena, such as single solitons, doubly solitons, multi-solitons, and lump solitons, as well as patterns resembling petals and peakons, are depicted. Subsequently, the bifurcations, sensitivity, and chaos dynamics of the discussed equation are examined. The dynamic behavior of the system is exhaustively analyzed through phase portraits and thorough bifurcation analysis. To further assess the stability of solutions, a sensitivity analysis of the dynamic model is performed using the Runge-Kutta method via MATLAB. The outcomes of this study enhance the understanding of nonlinear dynamics in various fields, including mathematical physics and fluid dynamics. The findings of this study are expected to be highly useful across a range of scientific disciplines and advanced research contexts.