This article uses Hirota's bilinear method (HBM) and appropriate transformations to investigate several lump solution forms in a cascaded system with spatiotemporal dispersion (STD) and Kerr law nonlinearity (KLN). The vector-coupled nonlinear Schrödinger equation is the mathematical model that describes how different solitons propagate through a cascaded system. Using the positive quadratic assumption in bilinear form, we evaluate lump solutions. By using the single and double exponential ansatz in bilinear form, respectively, we additionally investigate lump single-strip and double-strip soliton interactions. Furthermore, by using trigonometric and hyperbolic functions, respectively, we are able to find lump periodic and rogue wave solutions. Additionally, we discuss and illustrate the geometry of our solutions in multiple dimensions, such as contour plots and 3D. We also compute the stability of our solutions.

This research extensively investigates the solitary wave solutions of the fractional coupled Konno–Oono model, a prevalent framework in diverse scientific and engineering disciplines. Three mathematical methods called the simple equation method, the modified extended auxiliary equation mapping method, and the exponential-expansion method are gradually employed to derive the analytical solutions. Moreover, Mathematica 13.0 software is used to perform the analytical computations and graphical simulations. The explored outcomes have significant applications in the realm of magnetic fields. After the careful selection of parametric values under constrained conditions, some solutions are plotted in 2-dimensional and 3-dimensional spaces to understand the physical phenomena of the concerned model. Importantly, our findings affirm that the employed methods not only yield complete and uniform responses but also showcase simplicity, effectiveness, and remarkable computational efficiency. Hence, our research contributes valuable insights into the behavior of the fractional coupled Konno–Oono model, paving the way for enhanced comprehension and potential applications in magnetic field studies.

The emphasis of this work is to apply the complete discriminant system (CDS) of a polynomial method (CDSPM) to obtain soliton solutions for the extended Korteweg–de Vries equation (EKdVE). The model of the KdV equation is suitable for describing water waves with shallow water, where the water depth is up to the wavelength. We explore exact and analytical solutions such as the hyperbolic function (HF), the Jacobian elliptic function (JEF), solitary wave (SW) solutions, and trigonometric solutions. Additionally, we apply bifurcation concept in qualitative model of the EKdVE. The procedure involves transforming the system into a planer dynamical system through a given transformation and examining bifurcation analysis. We also explore phase portraits which assist in determining the stability of the equilibrium point of the system. Additionally, chaotic behaviour (CB) exhibits the exponential divergence of the orbits, i.e., tiny differences in the initial condition result in widely divergent orbits over a time period. To explore the potential of CB, we insert a perturbed term to the dynamical system and proceed with caution to explore the model. From these, we obtain a linear combination resulting in a strong dynamic mathematical structure. Finally, we apply Jupyter as a machine learning program as well as Mathematica to graph some obtained solutions in various dimensions such as three-dimensional (3D) and two-dimensional (2D) plots by using some packages of PYTHON such as numpy, scipy.integrate, and matplotlib.pyplot. Additionally, we explore energy balance method (EBM), presenting approximate periodic solution of non-linear oscillatory systems. The method is based on principle of conservation of energy, total energy (sum of kinetic energy as well as potential energy) is conserved.

Our study analyzes the two models of the nonlinear Schrödinger equation (NLSE) with polynomial law nonlinearity by powerful and comprehensible techniques, such as the variational principle method and the amplitude ansatz method. We will derive the functional integral and the Lagrangian of these equations, which illustrate the system's dynamic. The solutions of these models will be extracted by selecting the trial ansatz functions based on the Jost linear functions, which are continuous at all intervals. We start with the Jost function that has been approximated by a piecewise linear function with a single nontrivial variational parameter in three cases from a region of a rectangular box, then use this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by piecewise linear ansatz function in two cases of the two-box potential, then approximate it by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function, and finally, will be approximated by the tanh function. Also, we utilize the amplitude ansatz method to extract the new solitary wave solutions of the proposed equations that contain bright soliton, dark soliton, bright-dark solitary wave solutions, rational dark-bright solutions, and periodic solitary wave solutions. Furthermore, conditions for the stability of the solutions will be submitted. These answers are crucial in applied science and engineering and will be introduced through various graphs such as 2D, 3D, and contour plots.

Nonlinear stochastic models play a crucial role in describing complex wave phenomena in multidimensional physical systems. Motivated by this, we investigated the Stochastic Nizhnik-Novikov-Veselov (SNNV) equation to explore its solitary wave dynamics, chaotic behavior, bifurcation structures, sensitivity, and stability characteristics. We used the extended modified auxiliary equation mapping (EMAEM) method, an enhanced analytical framework with greater flexibility and broader solution structures versus conventional methods. With this approach, we derived new families of exact solitary wave solutions, including single, dark, and bright singular solitons. The proposed methods explain dynamical characteristics that were previously unexplored for the SNNV equation. The stochastic model will be converted into a dynamical system using the Galilean transformation. This approach enables exploration of its dynamical behavior via stochastic processes. We used Poincaré maps, phase portraits, and time-series trajectory simulations to establish its strong stochastic behavior, including chaotic behavior. To show that these methods are dynamically stable, we conducted a stability analysis using the Hamiltonian system framework. The proposed study will significantly advance the dynamic interpretation of chaos in the SNNV equation and establish the superiority of EMAEM approach for wave structure formation.

This research conducted a study of the nonlinear (3 + 1)-dimensional Vakhnenko-Parkes (VP) equation since it acts as a vital model for high-frequency wave perturbations in relaxing high-rate active barotropic media. The analytical solutions emerged through the modified auxiliary equation method and the improved F-expansion method which serve as advanced tools for studying nonlinear waves. These methods present diverse solutions that contain solitary waves combined with periodic waves and rational forms. The obtained solutions exhibit their behavior through illustrated 2D and 3D plots showing how waves evolve and how their structures transform with varying parameters. This visual analysis shows how solutions disperse and stay stable thus making them relevant for fluid dynamics investigations and wave propagation studies. The analytical understanding of nonlinear wave models in high-frequency barotropic systems receives new insights through our results which enhance mathematical and physical VP equation descriptions.

In this study, we apply the $(m+\frac{1}{{\mathrm{\Phi }}^{\prime }})$-expansion and modified extended tanh function (METHF) methods to investigate the exact solutions of the new Kairat-Ⅱ-Ⅹ model. Using these methods, new exact solutions are derived for the proposed model. The hyperbolic, periodic, and singular forms of exact solutions are among those obtained. The propagating behaviors of wave solutions are depicted using three-dimensional (3D), contour, and two-dimensional (2D) surfaces, providing comprehensive visualizations of the wave dynamics. Physical meanings of the chosen solutions are determined through these simulations. Traveling wave solutions of nonlinear partial differential equations can be obtained using the techniques presented in this paper since they have been shown to be dependable, strong, and effective. Furthermore, the phase portrait has been thoroughly analyzed according to the equilibrium points, and various scenarios have been visualized using these portraits. With the use of time series graphs, Poincaré maps, and 3D and 2D plots, the impact of the perturbation term has been thoroughly investigated. The system's periodic, quasi-periodic, and chaotic structures are clearly illustrated by these depictions. The results enhance the understanding of the new Kairat-Ⅱ-Ⅹ equation's dynamic structure and how it applies to real-world events.