A Generalized Template Framework for Constructing Exact Analytical Solutions to Various Nonlinear Partial Differential Equations

Abstract

This paper introduces a unified analytical framework for solving a broad class of nonlinear partial differential equations by mapping them onto a fundamental second-order nonlinear template equation. While various direct methods such as the G′/G-expansion , Kudryashov, and Jacobi elliptic function methods are widely used, we demonstrate that their underlying mechanism relies on reducing nonlinear partial differential equations to ordinary differential equations via traveling-wave transformations. By identifying a central prototype equation, y′′(ξ)=Ay2(ξ)+By(ξ), we provide a systematic approach to extract exact solutions for ten renowned nonlinear equations from diverse physical domains. Beyond algebraic derivations, the study investigates the structural and dynamical properties of the template equation, including phase-plane geometry, stability analysis, and a Poincaré-map and Lyapunov-exponent-verified route to deterministic chaos under periodic perturbations. The results confirm that the proposed template not only simplifies the solution-finding process but also serves as a robust tool for predicting complex transitions and stability regimes in nonlinear dispersive media.