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The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems.

In this paper, we investigate the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony equation using two effective methods: the unified scheme and the advanced auxiliary equation scheme, aiming to derive precise wave solutions. These solutions are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. Visual representations, including three-dimensional (3D) and two-dimensional (2D) combined charts, are provided for some of these solutions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through diverse figures, illustrating the profound impact of nonlinearity. Additionally, we briefly investigate the Hamiltonian function and the stability of the model using a planar dynamical system approach. This involves examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results demonstrate that these methods are reliable, straightforward, and potent tools for analyzing various nonlinear evolution equations found in physics, applied mathematics, and engineering.

The simplified modified Camassa-Holm (SMCH) equation is an important nonlinear model equation for identifying various wave phenomena in ocean engineering and science. The new auxiliary equation (NAE) method has been applied to the SMCH equation. Base on the method, we have obtained some novel analytical solutions such as hyperbolic, trigonometric, exponential, and rational function solutions of the SMCH equation. For appropriate values of parameters, three dimensional (3D) and two dimensional (2D) graphs are designed by Mathematica. The stability of the model is also discussed in this manuscript. The dynamic and physical behaviors of the solutions derived from the SMCH equation have been extensively discussed by these plots. All our solutions are indispensable for understanding the nonlinear phenomena of dispersive waves that are important in ocean engineering and science. In addition, our results are essential to clarify the various oceanographic applications containing ocean gravity waves, offshore rig in water, energy associated with a moving ocean wave and numerous other related phenomena. Finally, the obtained solutions are helpful for studying wave interactions in many new structures and high-dimensional models.

Abstract The doubly dispersive (DD) equation finds extensive utility across scientific and engineering domains. It stands as a significant nonlinear physical model elucidating nonlinear wave propagation within the elastic inhomogeneous Murnaghan’s rod (EIMR). With this in mind, we have focused on the integration of the DD model and the modified Khater (MK) method. Through the wave transformation, this model is effectively converted into an ordinary differential equation. In this paper, the goal of our work is to explore new wave solutions to the DD model by using the MK scheme. These solutions provide extremely helpful insights into the operation of the system. The three-dimensional (3D) plot and two-dimensional (2D) combined plot via the impacts of the parameters are provided for various parameters in this manuscript. We also discussed the dynamical properties of the model, which are accomplished through the bifurcation analysis, and also found the Hamiltonian function. This research makes a substantial contribution to the area by increasing our understanding of wave solutions in the DD, introducing novel investigation tools, and carrying out an in-depth investigation of the bifurcation and stability aspects of the system. As a direct result of this research, novel openings have been uncovered for further investigation and application in the various disciplines of science and engineering.

In this work, numerous nonlinear evolution equations, namely the nano-ionic current equation, have wide applications in various fields of science and technology. To achieve this, we employ the generalized auxiliary equation and extended hyperbolic function techniques to investigate the exact wave solutions of the model under consideration. We obtained a range of exact wave solutions, including bright, dark, singular-periodic, periodic, and w-shape solutions, for the nonlinear model under examination, as well as rational and exponential function solutions with various structures, which are explored in our manuscript. In this paper, we compared the solutions obtained using the GAE and EHF approaches and also compared the obtained solutions using the GAE scheme with the solutions reported in Aljahdaly et al. [43]. Moreover, the three-dimensional and contour graphs are included to illustrate the structure of some obtained solutions by selecting specific parameter values. Bifurcation analysis is illustrated by phase plane analysis; phase plots are employed to depict the dynamic structure and are also analyzed for the existence of other solutions. These techniques serve as reliable, simple, and potent tools to analyze various nonlinear evolution equations found in physics, applied mathematics, and engineering.

Nonlinear models (NLMs), being an important topic in mathematical physics, have attracted a lot of attention in the international research community because they have numerous uses in human life. These NLMs are typically implemented to illuminate various complicated natural phenomena. NLMs have been seen in fluid mechanics, optical physics, signal processing, plasma physics, acoustic waves, traffic flow, atomic physics, mathematical physics, nuclear energy, and other applications. That's why we consider the Mikhailov-Novikov-Wang (M-N-W) integrable equation. In this project, we acquire exact solutions (ESs) and numerical solutions (NSs) of the M-N-W integrable equation. Primarily, we use the modified (G′/G)-expansion method (MDG/GEM) and adomian decomposition method (ADM) to find new ESs and NSs of the M-N-W integrable equation. This present project work is mainly concentrated on finding new ESs and NSs for a broad class of the M-N-W integrable equation using the MDG/GEM and ADM, respectively, and making a comparison between these two kinds of solutions. Additionally, using recent scientific instruments, the two-dimensional, three-dimensional, and contour plots are displayed, as well as comparison plots between graphs of analytical and numerical solutions of the M-N-W integrable equation. NLMs have appeared in a wide variety of engineering, physics, and applied mathematics; hence, it is natural to expect that this project should have broad applicability.