The dynamic behavior of ion-acoustic waves (IAWs) inside a magnetized auroral zone plasma including ions as well as two-temperature electrons is investigated. The Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations are provided using the reductive perturbation method (RPT). In fluid dynamics, plasma physics, and electrodynamics, mKP equation can be utilized to characterize wave phenomena. It increases our interest due to its significant physical and mechanical value. The new extended direct algebraic approach is used to compute several traveling wave patterns for the nonlinear mKP equation. It is a simple, concise and general method in symbolic computations for solitary wave solutions for nonlinear problems. It yields dark solitary solutions, the community for semi-dark soliton results, dark singular soliton effects, kink soliton results, and soliton consequences encompassing trigonometric function, mixture hyperbolic function, as well as rational functions. Some arbitrarily chosen solutions are displayed visually for physical understanding. Moreover, the sensitivity of the perturbed dynamical structure equivalent to the mKP equation has been explored. Furthermore, the effect of IAWs spectral index on the parameters of the solitary-wave solutions and the sensitivity of the perturbed dynamical system of mKP equations is investigated in detail.

The investigation presented in this study delves into the analysis of Lie symmetries for the bistable Allen-Cahn (BAC) equation with a quartic potential, specifically applied to the biofilm model. By employing the Lie symmetry method, we have acquired the Lie infinitesimal generators for the considered model. Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. The obtained solutions manifest in various forms including dark, bright, kink, anti-kink, and periodic types using diverse strategies. To enhance the physical interpretation, the study presents 3D, 2D, and contour plots of the acquired solutions. Every graph’s wave-like structure contains information about the structural behaviour of the bacteria that build biofilms on surfaces where rectangles have different densities. This analysis enhances comprehension of the complex dynamics present in areas like fluid dynamics, fiber optics, biology, ocean physics, coastal engineering, and nonlinear complex physical systems.

In this paper, we mainly focus on studying the dynamical behaviour and soliton solution of the coupled Benjamin-Bona-Mahony-Korteweg–de Vries (BBM-KdV) system, which characterizes the propagation of long waves in weakly nonlinear dispersive media. The paper utilizes different tools to detect chaos, such as time series analysis, bifurcation diagrams, power spectra, phase portraits, Poincare maps, and Lyapunov exponents. This analysis helps in more accurate predictive modeling of the systems. This understanding can aid in the design of control strategies, resulting in enhancements in prediction, control, optimization, and design. Additionally, we construct the system’s solitary wave structures using the Jacobi elliptic function (JEF) method. We identify periodic wave solutions expressed in terms of rational, hyperbolic, and trigonometric functions. Certain parameter values can lead to periodic wave solutions, solitary waves (bell-shaped solitons), shock wave solutions (kink-shaped soliton solutions), and double periodic wave solutions.