The increasing complexity of nonlinear dynamical systems in modern science has generated an urgent need for analytical and semi analytical techniques capable of producing reliable and interpretable solutions without relying exclusively on heavy numerical simulations. Physical wave propagation, epidemiological processes, and population dynamics are all governed by nonlinear differential equations whose solutions determine prediction, stability, and control. Within this scientific context, the differential transform method has emerged as one of the most powerful series based analytical techniques for solving both linear and nonlinear differential equations across multiple domains. The present article develops a comprehensive, unified, and original analytical framework that integrates classical differential transform theory with its modern multi step, projected, adaptive, and hybrid extensions as reported in the literature.

Drawing strictly from the foundational and applied studies of Kangalgil and Ayaz, Ravi Kanth and Aruna, Hassan, Ayaz, Jang, Yildirim, Gokdogan, Odibat, and their collaborators, this study presents a fully synthesized theoretical structure that explains why and how differential transform based methods outperform traditional perturbation, decomposition, and purely numerical approaches for nonlinear models. The article does not merely summarize previous findings but instead reconstructs the conceptual logic that underlies them, tracing the epistemological roots of transform based series representations and explaining how they generate stability, convergence, and accuracy in nonlinear contexts.

A particular emphasis is placed on how differential transform techniques enable the extraction of physically meaningful behaviors from nonlinear partial differential equations such as the Korteweg de Vries and modified Korteweg de Vries equations, which describe solitary wave propagation in fluid and plasma physics. Through detailed theoretical analysis, the article demonstrates how differential transform solutions preserve wave coherence, soliton structure, and nonlinear interactions in a way that conventional discretization schemes often fail to capture. These theoretical insights are directly connected to the work of Kangalgil and Ayaz, who showed that differential transform based solutions reproduce solitary wave profiles with remarkable precision when compared with exact analytical solutions.