Bifurcation analysis, and exact solutions of the two-mode Cahn–Allen equation by a novel variable coefficient auxiliary equation method

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This document elaborates on a newly introduced analytical method known as the “Variable Coefficient Generalized Abel Equation Method,” as proposed by Hashemi in Hashemi (2024), designed specifically for addressing the two-mode Cahn–Allen equation. Diverging from conventional techniques that heavily rely on constant coefficient ordinary differential equations and auxiliary ordinary differential equations, our method innovatively incorporates variable coefficient ordinary differential equations within a sub-equation framework. Demonstrating its versatility, we apply this innovative technique to the two-mode Cahn–Allen equation, showcasing its effectiveness and efficiency through the derivation of analytical solutions. Notably, this method emerges as a promising tool for tackling complex nonlinear partial differential equations prevalent in fluid dynamics and wave propagation scenarios. Beyond merely expanding the repertoire of available analytical tools, our approach contributes to advancing solutions for various models within the realm of mathematical physics. Various forms of exact solutions, including exponential-type solutions, Kink solitons, dark solitons, and bright soliton solutions, are obtained for the model under consideration. Moreover, we delve into the analysis of bifurcation, chaotic behavior, and sensitivity within the context of the two-mode Cahn–Allen model, further enhancing the depth and breadth of our study. Three equilibria are analyzed across various classifications, including center point, focus point, saddle point, and node point. Chaotic behavior of the corresponding dynamical system is considered by adding the function ω1sin(ω2ζ). Lastly, sensitivity analysis of the system is conducted by examining different parameters of the model and imposing noise to the initial conditions.