In this study, we examined the nonlinear dynamics of the Boussinesq equation, a foundational equation in ocean engineering to model and investigate the behavior of waves in shallow water. The novel (G'/G²)-expansion method was employed to obtain different soliton solutions, including periodic, bright, W-type, and bell-shaped soliton solutions. These solutions are illustrated through 2D, 3D, and contour plots. We discovered different dynamical behavior, including periodic, quasi-periodic, and weak chaos, depending on the choice of initial conditions and parameters. The important outcomes included the detection of multistable attractors and the presence of weak chaotic behavior supported by Lyapunov exponents. These understandings have important effects in practical uses such as energy harvesting and wave control in ocean systems, where handling and understanding system transitions and stability is crucial. These findings also give a framework for further examination of stability and control in nonlinear wave systems.

Viscous dissipation and thermal radiation are essential in governing the dynamics of boundary layer flow, particularly in high-temperature engineering systems such as gas turbines, combustion engines, and industrial furnaces. Thermal radiation emerges as a primary factor of heat transfer under such conditions, while viscous dissipation contributes to the transformation of kinetic energy into thermal energy through internal frictional forces within the fluid. The present study investigates the three dimensional magneto-hydrodynamic (MHD) flow of a radiative Eyring-Powell nanofluid towards a stretchable, porous surface. The governing equations, initially formulated as partial differential equations (PDEs), are reduced to a set of coupled ordinary differential equations (ODEs) through similarity transformations. These transformed equations are numerically solved using MATLAB bvp5c solver. A detailed parametric analysis is performed to examine the impact of key dimensionless quantities, including slip parameter, Lewis number, Eckert number, thermophoresis parameter, magnetic field strength and Brownian motion parameter, on velocity profile, temperature profile and concentration distributions. The analysis reveals that the fluid velocity decreases with an increase in the magnetic field strength, whereas it exhibits an increasing trend with higher values of the Eyring–Powell fluid parameter. This paper convers the following key points: • Modeled the dynamical flow equation for hybrid Eyring-Powell nanofluid. • Analyzed the magnetic force impact on the velocity curve. • Graphical interpretations are presented, highlighting the effects of physical parameters.

This work investigated the Kadomtsev Petviashvili-modified equal width (KP-mEW) equation describing ocean waves. Our focus was on the analysis of the KP-mEW equation from various angles, including the study of soliton solutions, bifurcation analysis, multistability, and Lyapunov exponents. First, a transformation was used to transform the partial differential equation (PDE) into an ordinary differential equation (ODE), from which the soliton solutions were obtained by using a new modified (G'/G²)-expansion method. We investigated different types of solutions of the KP-mEW equation with various parameters, including kink, periodic, singular periodic, singular kink, and singular periodic-kink solutions. We also simulated 3D and 2D plots for some solutions to enhance the visualizations. These graphical representations provide important information about the patterns and dynamics of the solutions, leading to a strong understanding of the behavior and applicability of the model. We also observed the chaotic behavior of the system by adding a perturbation term and analyzed the chaotic behavior through bifurcation plots, multistability and time series analysis, and Lyapunov exponents and obtained various dynamic modes such as periodic and quasi-periodic types. A comparison of the obtained solutions with the existing solutions was also presented in the form of Table 1.