Chaotic analysis and a damped oscillator solitary wave structures to the generalized reaction Duffing model

Abstract

The aim of this research is to obtain soliton solutions for the generalized reaction Duffing model, a framework that generalizes many important models that illustrate key phenomena in science and engineering. In contrast to regular harmonic motion, this equation describes the motion of a damped oscillator with a more complex potential. We used the Kumar–Malik method in this work to obtain analytical solutions for the generalized reaction Duffing model, which is the first time this method has been used to extract soliton solutions in this particular setting. The equation is first reformulated as a nonlinear ordinary differential equation using traveling wave transformation. The approach proves particularly effective in handling nonlinear partial differential equations, yielding hyperbolic, Jacobi elliptic, trigonometric, and exponential function solutions under appropriate parameter constraints. A variety of innovative solutions emerge, including periodic wave solutions, dark compacton waves, kink waves, singular kink waves, bright solitons, breather waves, and singular-shaped solitons via the Kumar–Malik method. The solutions are then shown visually to demonstrate the wave behavior under various conditions. Our findings enhance the comprehension of the Duffing equation’s behavior across different physical contexts. The research uses extensive 2D and 3D graphic plot solutions of the proposed solutions for a better graphical understanding of the physical perimeters of solutions and proves the feasibility of the proposed method in solving complex nonlinear equations. The Chaotic analysis has also been discussed by perturbation term and initial conditions. It is important to note that the proposed methods are competent, credible, and interesting analytical tools for solving nonlinear partial differential equations. In addition, these solutions represent a valuable resource for the understanding of the complex behavior of physical systems, as well as for inspiring future research.