Exact and numerical investigation of soliton dynamics in nonlinear oscillatory system

The generalized reaction Duffing model is an important nonlinear system that describes a variety of oscillatory phenomena arising in physics, engineering, and applied sciences. In this work, we investigate its rich soliton dynamics through a combined analytical and numerical framework. Exact traveling-wave solutions are constructed by employing the generalized Riccati equation mapping method (GREMM), whereby the governing partial differential equation is transformed into a nonlinear ordinary differential equation via a traveling-wave transformation. Under suitable parameter constraints, several families of exact solutions expressed in hyperbolic, trigonometric, and exponential functions are obtained, yielding dark, bright, singular, dark-bright, dark-singular, and periodic-singular soliton structures. The validity of all derived solutions is verified through direct substitution into the original governing equation. To complement the analytical findings, the differential transform method (DTM) is utilized to numerically approximate selected solution families, and an excellent agreement between the exact and differential transform method solutions confirms the accuracy and effectiveness of the proposed semi-analytical technique. A conserved-quantity-based stability criterion is further employed to examine the robustness of the obtained solitary waves, revealing that the considered solutions remain stable for representative parameter values. Moreover, comprehensive graphical illustrations, including three-dimensional absolute-value surfaces, density distributions, and two-dimensional wave profiles, provide physical insights into the localization, propagation, and interaction characteristics of the nonlinear wave structures. The present study demonstrates the capability of the combined generalized Riccati equation mapping method and differential transform method framework to analyze complex nonlinear dynamical systems and offers benchmark solutions that may be useful for future investigations of generalized Duffing-type models and nonlinear oscillatory media.