Exact Traveling Wave Solutions of a Fractional Nonlinear Korteweg–de Vries Equation With Spatiotemporal Interaction Terms via the Atangana–Baleanu Operator

This study investigates the dynamical behavior of a nonlinear Korteweg–de Vries (KdV) equation incorporating second-order spatiotemporal interaction terms within the framework of the Atangana–Baleanu fractional operator. The proposed model provides a generalized mathematical description of nonlinear wave propagation in dispersive media with memory effects. The extended hyperbolic function method is employed to construct a variety of exact traveling wave solutions, including bright solitons, dark solitons, singular solitons, periodic waves, and mixed nonlinear wave structures expressed in hyperbolic and trigonometric forms. The obtained solutions exhibit diverse dynamical behaviors influenced by the fractional order and physical parameters of the system. To further illustrate the physical relevance of the results, two- and three-dimensional graphical simulations are presented. The analysis shows that the Atangana–Baleanu fractional operator significantly affects the amplitude, localization, and propagation characteristics of nonlinear waves. The proposed model, therefore, provides a useful framework for understanding nonlinear wave dynamics in physical systems with memory effects, such as shallow water waves, plasma physics, and nonlinear transmission media.