Dynamical analysis, synchronization of chaos, and soliton solutions of a nonlinear (3+1)-dimensional extension of KdV equation with second-order time-derivative

The investigation of advanced versions of nonlinear evolution equations is crucial for demonstrating complex wave phenomena in different physical systems. Extension and generalizing the classical KdV equation, specifically those involving higher-order derivatives and nonlinearities, provide indepth insights into rich dynamical behaviors such as chaos dynamics, synchronization and solitons. Studying these features of higher order equations not only broadens the theoretical foundation of nonlinear science but also paves the way for practical applications in fields ranging from fluid dynamics to secure communications. In this paper, a comprehensive dynamical and analytical investigation of a generalized version of KdV having a second order temporal derivative term is presented. Using wave transformation, we convert the KdV equation into a system of two ODEs. Detailed investigation of the dynamics of the system such as bifurcation behavior, chaotic dynamics, multi-stability, power spectra, and recurrence plots are portrayed. A statistical analysis of the chaotic dynamics is demonstrated via the probability density of the state variables. Synchronization of the chaotic system and its properties are examined with particular attention given to synchronization error, phase space comparisons, and cross-correlation analysis. Analytical investigation is carried out by utilizing the Kumar–Malik method. Several novel soliton solutions of the advanced version of KdV equation is constructed in terms of Jacobi-elliptic, trigonometric, and hyperbolic functions. Also, the linear stability of the equation is established, providing valuable insights into the stability of the proposed KdV equation. The results are presented graphically in 3D and 2D figures, collectively enhance the understanding of the complex behaviors exhibited by the extended KdV equation.