Chaotic response, multistability and new wave structures for the generalized coupled Whitham–Broer–Kaup–Boussinesq–Kupershmidt system with a novel methodology

Nonlinear science constitutes a pivotal domain of scientific research, focusing on the investigation of inherent characteristics and common attributes of nonlinear phenomena. In this work, we present the nonlinear aspects of the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system exploring the attributes of dispersive long waves in relation to shallow oceanic settings. By employing a new generalized exponential differential function approach, we successfully derive a variety of new structures, particularly lump-type, lump-periodic, multi-peakon, hybrid lump-dark and lump-bright solutions. These solutions are fundamental in illustrating the rich structure and diverse dynamics inherent in nonlinear higher-dimensional systems. We present these solutions graphically in 3D, contour and density plots to gain a comprehensive insights. In addition to this, we explore the nonlinear characteristics of a perturbed dynamical system to identify the chaotic response by using the idea of chaos theory. Chaotic phenomena is observed and confirmed by adopting different chaos detection tools. Also, we perform the multistability analysis of the perturbed dynamical system under varying initial conditions. This analysis demonstrates that even minor changes in the ICs can lead to shifts in the system’s behavior, transitioning from a stable to an unstable state. Meanwhile, this work represents a novel and significant contribution to the study of the system, enhancing our understanding of localized waves and their dynamics. It also aids in predicting and managing the impact of perturbations in real-world applications such as climate models and engineering systems.