Dynamical behavior of kink solitons in nonlinear Chaffee-Infante equations with chaotic and bifurcation analysis

Abstract

The Chaffee-Infante Equations (CIEs) are modified types of reaction-diffusion equations which are frequently employed in research of phase transitions, pattern generation and nonlinear wave dynamics. The main purpose of this work is to focus on building and analyzing soliton solutions for (1+1)- and (2+1)-dimensional CIEs through an analytical method known as $\left(\frac{G^{\prime }}{G}\right)$-expansion method. The strategic $\left(\frac{G^{\prime }}{G}\right)$-expansion technique first converts CIEs into Nonlinear Ordinary Differential Equation (NODEs) using wave transformations which are subsequently transformed into systems of nonlinear algebraic equations under the supposition of closed-form solutions. The solutions of the resulted systems yield numerous soliton solutions in the form of rational, exponential, trigonometric and hyperbolic functions when analyzed by using the Maple tool. Some soliton solutions are assessed through illustrated contour and 3D visualisations for specified parameter values to confirm the existence of kink soliton solutions such as cuspon, anti-kink, bright, dark, dark-bright and multiple kink solitons in CIEs. The chaotic behavior of the perturbed dynamical systems is also investigated through Gillian transformation method and time series method, noting its existence in the dynamical system that has been perturbed and getting favorable outcomes about the chaotic behaviors of CIEs. Moreover, the research shows that $\left(\frac{G^{\prime }}{G}\right)$-expansion method works as an efficient robust simple method which generates numerous soliton solutions applicable to various Nonlinear Partial Differential Equations (NPDEs) in mathematical sciences.