Exact Soliton Solutions and Lyapunov-Based Nonlinear Dynamical Analysis of a Classical Kolmogorov–Petrovsky–Piskunov Equation

Abstract

In this study, an analytical and dynamical study of the classical Kolmogorov–Petrovsky– Piskunov (KPP) equation is carried out by means of the improved generalized Riccati equation mapping (IGREM) method. An appropriate traveling-wave transformation is used to reduce the governing equation and several families of exact analytical solutions are systematically built, including hyperbolic, trigonometric, rational, and exponential solutions. The solutions are different nonlinear wave structures that are shown to be useful for the IGREM framework of nonlinear evolution equations. In addition to the exact solution construction, qualitative dynamics of the reduced system are explored using phase portraits, bifurcation analysis, periodic perturbations, Lyapunov exponents, chaotic dynamics, and multistability. The Lyapunov analysis gives a quantitative description of sensitivity to initial conditions, and the presence of two different attractors means that the system is multi-stable. The results show that the parameters of the system and the external periodic perturbations can lead to transitions between regular and complex dynamical regimes. The present study is essentially mathematical in nature, but the wave structures and the nonlinear dynamical features that have been obtained could be of theoretical interest in various contexts of nonlinear wave phenomena in the context of microwave propagation under appropriate modeling assumptions. The proposed framework therefore provides a unified and systematic approach for investigating both exact traveling-wave solutions and complex dynamical behaviors in nonlinear reaction–diffusion systems.