Chaotic behavior, sensitive analysis and dynamics of invariant formulation of nonclassical symmetries to the nonlinear Tzitzéica equation arising in mathematical physics

This dissertation provides a comprehensive nonclassical symmetry analysis of the Tzitzéica-type nonlinear evolution model, which is a prominent model that arises in quantum field theory and nonlinear optics. The analysis is carried out in two main phases. In the initial phase, systems of nonlinear partial differential determining equations are developed, tailored to the dimensional configuration of the model. Two different cases of these systems are constructed and meticulously solved, leading to the identification of numerous new nonclassical symmetries. Secondly, these nonclassical symmetries are used to categorize the unknown functions and to extract their associating invariant forms. Several novel explicit and accurate solutions of the model are obtained using these categorized functions. To emphasize these solutions’ salient characteristics and possible applications the dissertation also provides graphical representations of them. Additionally, the study uses the Galilean transformation to examine the model’s sensitivity and chaotic behavior. To explore the intricate and elusive character of chaos that is inherent in the system, time series, Lyapunov exponents, and power spectra are used as effective tools. Together with the insights into the dynamic behavior of the model, these exact and explicit answers are essential for expanding knowledge across a number of applied science fields.