Bifurcation Analysis, Classification of~Traveling Wave Solutions, and Perturbation-Induced Chaotic Behaviors of~the Extended (3+1)-Dimensional Nonlinear Kudryashov’s Equation

This paper conducts bifurcation analysis of the extended (3+1)-dimensional nonlinear Kudryashov’s equation (EKE), classifies traveling wave solutions, and investigates the chaotic behaviors of the equations under specific perturbations. Firstly, the traveling wave transformation is introduced to reduce the EKE to a dynamical system, and then Gaussian solitons are obtained using the generalized trial equation method. In the following step, the existence of periodic and soliton solutions is confirmed through qualitative analysis provided by bifurcation theory and phase diagrams. To verify the conclusions of the qualitative analysis, the complete discrimination system for the polynomial method (CDSPM) is employed for classification, thus enabling the traveling wave solutions. Finally, sinusoidal and Gaussian perturbations are introduced, and the corresponding Lyapunov exponents are used to verify the existence of chaotic behaviors. This paper presents the first application of the CDSPM method to the EKE, yielding several novel results including Gaussian soliton solutions, prior estimates for periodic and soliton solutions. Additionally, traveling wave solutions in the form of hyper-elliptic functions and inverse trigonometric functions are also initially given. Finally, the chaotic behavior of the EKE under perturbation terms is presented. Abstract Published by the Jagiellonian University 2026 authors