Exploring chaos, multistability, and interaction patterns in (3+1)-dimensional KdV-BBM model

This paper examines the exact solutions of the (3+1)-dimensional Korteweg–de Vries Benjamin–Bona–Mahony model. The given model depicts the two-way transmission of waves and describes large surface gravity waves with minimal amplitude. By employing the Hirota bilinear technique, we create a bilinear Bäcklund transformation with three equations and four free parameters. Based on the Bäcklund transformation, exponential and rational traveling wave solutions are calculated. Next, we successfully identify the lump interaction with different waves, two wave, and multiwave solutions by selecting the function in Hirota bilinear form as the general quadratic function, trigonometric function, and exponential function together with a suitable set of parameters. In order to identify the behavior of the waves, we also evaluated certain 3D, 2D, and contour profiles. Furthermore, we use the ideas of stability analysis and chaos theory to comprehend the planar dynamical system. The chaotic trajectory in the perturbed system is identified using phase portraits, time-scale graphs, Poincaré maps, bifurcation diagram, and Lyapunov exponent graphs. Additionally, a multistability study of the model is discussed for different starting conditions. The outcomes demonstrate how these methods are reliable, simple to use, and efficient for examining a range of nonlinear models that are present in mathematical physics and engineering fields.