The time-regularized long-wave equation is pivotal in understanding diverse wave dynamics, such as shallow water waves, pressure waves in liquids and gas bubbles, ion-acoustic waves in plasma, and nonlinear transverse waves in magnetohydrodynamics. The analysis is initiated by deriving the infinitesimal generators of the Lie group symmetries, followed by constructing a commutator table and adjoint table to study the algebraic structure of the equation’s symmetries. Using this symmetries, the time-regularized long-wave equation is systematically reduced and solved for invariant solutions associated with each symmetry. These solutions give insight into the inherent properties of the system and wave behavior. The extended hyperbolic function method is employed to derive exact solutions to the simplified versions of the time-regularized long-wave equation. This method enhances the solution process by generating soliton solutions including dark, singular, bright and periodic-singular solutions. To illustrate their behavior of obtained solutions, graphical representations are given by using different values of parameters. A comprehensive bifurcation analysis is also conducted to explore the qualitative changes in the dynamics of system, while the Hamiltonian structure of the equation is constructed to investigate its conservation properties. The study also investigates the phase portraits of the system to offer a visual interpretation of the solution trajectories in phase space. Finally, the modulation instability of the time-regularized long-wave equation through linear stability analysis is analyzed to provide a clearer understanding of the conditions that lead to the onset of instability in wave propagation.

Ultrasound imaging stands as a cornerstone of modern medical diagnostics, revolutionising clinical practice with its non-invasive, real-time visualisation of internal structures. Central to this technique is the propagation of ultrasound waves and their intricate interplay with biological tissues, culminating in the generation of intricate and detailed images. This study delves into the symbiotic relationship between solitons and ultrasound imaging within the framework of the Westervelt equation, a fundamental model governing ultrasound propagation. Employing the generalised Riccati equation mapping method and the generalised exponential rational function method, a diverse array of soliton solutions is elucidated, encompassing dark, kink, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Visualisation of these solutions through 3D plots, contour plots, and 2D plots at varying time intervals offers a captivating insight into their dynamic nature. We provide a comparison of these solutions through 2D plots at different parameter values, highlighting their varying impacts. Central to this study is the exploration of how these soliton solutions can be harnessed to enhance the quality and accuracy of ultrasound images in medical imaging. Through meticulous analysis of their characteristics, this research seeks to illuminate their potential applications, paving the way for a new era of precision diagnostics in healthcare. By conducting thorough mathematical analyses and numerical simulations, we seek to elucidate the complex relationship between soliton theory and ultrasound imaging, connecting the theoretical aspects of nonlinear wave phenomena with their practical applications in medical diagnostics. An intensive literature review underscores the novelty of our work.

The model is noteworthy for its nonlinearity and ability to describe the propagation of nonlinear waves inside the elastic, inhomogeneous Murnaghan’s rod equation. The fractional nonlinear doubly dispersive equation is applied in physics to model wave propagation in elastic materials and plasma, used to describe seismic wave dynamics and nonlinear effects in quantum systems. We used the β-fractional and M-truncated fractional derivatives to solve the fractional version of Murnaghan’s rod equation. The generalized Riccati equation mapping method has been utilized to find some new exact traveling wave solutions to the space–time fractional nonlinear doubly dispersive equation. It was observed that the singularity of water waves is influenced by the velocity and wave number parameters of soliton waves. To visually represent the solitons and categorize them, we utilized graphs, which revealed a diverse range of wave patterns that undergo changes corresponding to different values of σ and λ. In this study, we also added the analysis of chaotic behavior for the governed equation. By varying some parameters, the system’s transition from chaotic to quasi-periodic behavior is explored which highlights the impact of δ and φ on the systems dynamics. This study also investigates the phenomenon of multistability in a perturbed dynamical system, where multiple distinct behaviors, such as periodicity, quasi-periodicity and chaos, can coexist under the same set of parameters but different initial conditions. Through a detailed analysis of the system’s dynamics, we explored how varying initial constraints impact the evolution of the system.

In this article, we investigate the mystery of matter dominance over antimatter through gravitational baryogenesis in context of Einstein-(Ether gravity. This study aims to focus on the phenomenon of baryon asymmetry produced in the universe a while after the Big Bang. The supremacy of matter over antimatter is quantified the ratio and compared our outcomes with the observational bounds. We found this ratio is consistent with the observational data. For a further verification, we examine the Hubble parameter () for each model and compared them with observational values, finding a high degree of agreement. Additionally, we perform Chi-square (2) test on Hubble parameter for each model to assess their compatibility with recent observational data. Furthermore, we compared our findings with the Lambda CDM model and the latest Pantheon+SH0ES observational data. This comparison unveiled a reasonable level of consistency among the considered models and the observational data.

Investigating different types of solitons in heat transport presents promising opportunities for information processing technologies. In this paper, the heat conduction equation is analyzed using Lie symmetry analysis. We compute the optimal system of one-dimensional subalgebras for the heat conduction equation and utilize it to identify a set of invariant solutions. Additionally, we employ the Lie group method to reduce the heat conduction equation to new differential equations. Furthermore, we apply the generalized Riccati equation mapping method to extract solutions in the forms of kink, singular, dark-singular, and bright-kink solitons. We present our findings through 3D, density, 2D, and contour plots. In this paper, we extend our investigation to include modulation instability and gain spectrum analysis within the framework of nonlinear dynamical systems. Although the idea of using heat solitons for information transfer is still largely theoretical, it opens up an exciting area for further exploration. Studying these solitons through the lens of information theory provides new motivation for research, emphasizing their potential contribution to the development of heat-driven information systems. For novelty, we conducted a detailed comparison of our solutions with existing methods, highlighting the advantages and unique features of our approach.

A microcrystalline material refers to a crystallized substance or rock comprised of tiny crystals that can only be observed under a microscope. The strain wave equation is a fourth-order nonlinear partial differential equation encountered in the examination of non-dissipative strain wave propagation within microstructured solids. In this paper, the transmission of waves in microcrystalline materials is dictated by the non-dissipative case of strain wave equation's structure, accounting for multiple dimensions within microcrystalline structures. The simplest equation method is employed to extract multi-soliton solutions, while the modified Sardar subequation method is applied to identify additional soliton solutions, including bright, combined dark-bright, combined dark-singular, periodic singular, and singular solitons. Furthermore, the dynamical system bifurcation theory approach is utilized to investigate the phase diagrams of the governing equation. Further elaboration on the physical dynamical representation of the presented solutions is provided through profile illustrations. A comparison with the existing literature is also provided, highlighting the efficacy of our work. The significance of the acquired outcomes lies in their capacity to portray a wide array of intricate and diverse phenomena observed in both mathematical and physical systems.

This study explores optical solitons in the Biswas–Arshed equation within birefringent fibers. Employing the unified solver method, the ( 1/φ (η ),φ '(η )/φ (η )) method, and new Kudryashov’s method, we extract various optical soliton solutions, encompassing dark, singular, bright, and periodic forms. These solutions deepen our understanding of dynamic phenomena in birefringent fibers, showcasing their potential practical applications. The results, effectively visualized in 3D and 2D plots, reveal intricate patterns. Our research underscores the efficacy and simplicity of these approaches in obtaining optical solitons for diverse nonlinear evolution equations. The novelty lies in the advanced methodologies applied to investigate the Biswas–Arshed equation, yielding a diverse array of soliton solutions and their practical implications. This study not only presents a variety of solutions but also highlights their applicability across disciplines and real-world scenarios. Consequently, our research significantly contributes to advancing our understanding of optical solitons in birefringent fibers, offering a methodological breakthrough in engineering and applied physics.

In this study, we examine the dynamic behavior of optical solitons in the nonlinear conformable Gross–Pitaevskii equation within Bose–Einstein condensates, which refers to the phenomenon of many ultra-cold bosonic particles occupying a single quantum state. It is akin to the Ginzburg–Landau equation and resembles the nonlinear Schrödinger equation. We also discuss how fractional-order parameters affect the solution dynamics. In fiber optics, this model describes how light propagates in optical fibers when ultra-short pulses are generated. We characterize the model using the conformable time-fractional derivative operator to provide a more detailed description of the physical component. To extract exact solutions of the model, we employ the unified solver method, new Kudryashov method and ( 1/φ (ϱ ),φ ^'(ϱ )/φ (ϱ )) method. These methods for resolving nonlinear partial differential equations that arise in the natural sciences are intuitive, sturdy, and robust. Through consideration of the arising constraints over the parameters of the methods and nonlinear fractional Gross–Pitaevskii equation, a novel entire family of optical solitons are determined in the form of hyperbolic, trigonometric and rational function solutions. These solutions comprehend dark, bright, singular, and periodic-singular solitons. Visual representations of the derived solutions are provided through two-dimensional (2D) and three-dimensional (3D) graphs.

The (2 + 1)-dimensional Konopelchenko-Dubrovsky system contributes to the field of atmospheric science by investigating the behaviour of nonlinear waves, revealing subtle scattering effects and extended-range interactions within the tropical and mid-latitude troposphere. This equation provides insights into the interplay between equatorial and mid-latitude Rossby waves, capturing their complex interactions and dynamics. Nonlinear waves are significant in atmospheric processes and understanding their dynamics is important for comprehending weather patterns. This study centres around the utilisation of the simplest equation method and extended hyperbolic function method to analyse these waves and derives many solutions that illustrate different wave patterns and behaviours intrinsic to the governed system. The simplest equation method enables the derivation of kink and multi-soliton solutions. This method allows for the extraction of solutions characterised by multiple solitary waves propagating without distortion. On the other hand, the extended hyperbolic function method provides anti-bell-shaped, periodic, and singular solutions. These diverse solution types portray various wave patterns and behaviours intrinsic to the equation under study. Additionally, 3D and 2D graphical representations are generated to visually depict the obtained solutions. Hence, this study not only contributes to the comprehension of non-linear wave dynamics in atmospheric science but also imparts knowledge on the broader applicability of mathematical methods in uncovering the underlying intricacies of natural phenomena in different fields of non-linear sciences.

The main objective of this research is to investigate various important soliton solutions for long-short-wave interaction model (LSWI) through the application of the new extended direct algebraic methodology. Thorough investigation and accurate verification, of hyperbolic, single and periodic, mixed-wave solutions as well as mixed periodic, shock soliton, different complex combination solutions, mixed trigonometric solutions, trigonometric solutions, shock results, singular solution, mixed singular results, mixed complex solitary wave outcomes, along with mixed shock singular solution as well as the mixed trigonometric solution are discovered. Advanced solitary wave solitons are generated by modifying the values of the parameters implicated in the derived solutions. The importance of these solitons in the model is illustrated via contour plots, density plots, 2D and 3D visualizations. Additionally, the dynamical investigation is established, and then sensitivity analysis, as well as bifurcation analysis, are illustrations to depict various aspects. The obtained bifurcation findings show the dynamic behavior of long-short-wave interaction model from a geometric perspective. Transmission of optical solitons via nonlinear optics is provided by the obtained results. The wave dynamics of the solutions are shown to provide a more physical viewpoint on the outcomes, helping the reader to better understanding the nonlinear wave equation that represents physical processes.

The nonlinear Schr & ouml;dinger equation, held in high regard in the realms of plasma physics, fluid mechanics, and nonlinear optics, reverberates deeply within the field of ocean engineering, imparting profound insights across a plethora of phenomena. This article endeavours to establish a connection between the equation's theoretical framework and its practical applications in ocean engineering, presenting a range of solutions tailored to grasp the intricacies of water wave propagation. By employing three methodologies, namely, the simplest equation method, the ratio technique, and the modified extended tanh-function method, we delineate various wave typologies, encompassing solitons and periodic manifestations. Enhanced by visual representations, our findings have the potential to deepen the comprehension of wave dynamics, with promising implications for the advancement of ocean engineering technologies and the refinement of marine architectural design.

Comprehending the temperature-dependent variable viscosity Phan Thien Tanner fluid film flow over a horizontal heated plate with surface tension is essential to enhancing coating and lubrication process prediction models. This paper accords with the flow analysis of a temperature-dependent variable viscosity Phan Thien Tanner fluid film over a horizontal heated plate. The flow over a heated plate occurs as a result of the plate’s motion and the surface tension gradient. The Adomian decomposition method is utilized to solve a system of linear and nonlinear ordinary differential equations, resulting in series-form solutions. The series-form expressions for flow variables such as velocity, temperature, volume flow rate, and surface tension are derived. Moreover, the positions of stationary points are computed using MATHEMATICA. The analysis delineated that when the inverse capillary number, variable viscosity parameter, Deborah number, elongation parameter, and Brinkman number increase, the stationary points move closer to the heated plate. The temperature also rises with an increase in these parameters. The temperature rises with increasing viscous dissipation while it lowers with increasing thermal diffusion. When the Deborah number is high, the Phan Thien Tanner fluid behaves like a solid and the flow is only driven by the motion of the plate. A comparison between Newtonian and Phan Thien Tanner fluids is made for velocity, temperature, and stationary points as a special case.

Analyzing Phan-Thien Tanner fluid film behavior on a vertically upward moving tube can help predictive models in engineering, notably in coating and lubrication operations. This paper provides a theoretical analysis of the dynamics of stagnant rings and uniform Phan-Thien Tanner fluid film adhered to a vertically upward moving tube. Formulated ordinary differential equations are solved to get exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant rings. Highly nonlinear algebraic equations are solved using Newton's method in MAPLE to find the linear and exponential Phan-Thien Tanner film thicknesses. The uniform film thickness widens with increasing constant tube velocity, while it decreases with increasing Deborah number, Stokes number, and elongation parameter. The analysis delineates that stagnant rings shrink around the tube as the Stokes number, Deborah number, and elongation parameter increase. The minimal Stokes condition for the existence of a realistic stagnant ring is also determined. It has been established that whenever the Stokes number is less than the minimal Stokes condition, stagnant rings do not form. A comparison between the exponential Phan-Thien Tanner, linear Phan-Thien Tanner, upper convected Maxwell, and Newtonian fluids is also provided for stagnant rings and fluid film thickness. Following the application of an efficient approximation on tube geometry, plate geometry is approximated, and the outcomes are consistent with existing literature. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design.

The nonlinear Schrödinger equation is used to model various phenomena, such as solitons self-focusing effects and rogue waves. In the ocean engineering, the modified nonlinear Schrödinger equation investigates the behavior of water waves, considering the complex interaction of dispersion nonlinearity, and dissipation effects. By introducing fractional derivatives to the model, the M-fractional conformable modified nonlinear Schrödinger equation allows for the investigation of fractional order effects, which can study more accurately the behavior of wave propagation in real-world ocean engineering. The novelty of our research lies in the application of of the M-fractional conformable derivative on the governed equation which represents an advancement in the existing work, which have used nonlinear Schrödinger equations without fractional derivatives. Two powerful techniques: the Jacobi elliptic function method and unified solver method are applied to attain solutions to the M-fractional modified nonlinear Schrödinger equation. The several results, including dark, bright, singular, periodic, and dark-bright soliton solutions are obtained which provide valuable insights into the complex behavior of water waves in ocean engineering. Additionally, 3D and contour graphs have been provided to visually illustrate the impact of the fractional order. We also illustrate these solutions at different values of the fractional order which explain how variations in this parameter affect wave propagation. These findings will contribute to the advancement of ocean engineering techniques, enhancing our ability to design and implement effective solutions for coastal protection, offshore structures, and marine renewable energy systems.

In this study, the extended hyperbolic function method and unified solver method are applied to examine the propagation of optical solitons in perturbed nonlinear Schrodinger equation. The proposed techniques are characterized as simpler, more concise, and straightforward. They allow for the extension of the class of solutions and the retrieval of a diverse range of optical solitons, including singular, periodic singular, bright, and dark ones. This highlights the significance of the results. The authors emphasize the novelty of these techniques, as they have not been previously applied to recover optical solitons in this model These out comes may be beneficial for further understanding of numerous phenomena that arises in various physical systems and can be castoff for the optical communication resolution. The existence measures for these solutions are also unveiled in the form of constraint parameters. In addition, the pictorial representations of some solutions are depicted in figures 1–7 assuredly play a key part to understand the behavior and apprehending some of the physical features of the under consideration model. This combination of problem and methodologies has led to the discovery of a plethora of novel soliton solutions and their accompanying behaviors.

This article studies the reduced Ostrovsky equation by using Lie group theory, an efficient method for deriving symmetries in nonlinear equations. The method starts by finding infinitesimal generators and then adjoint and commutator tables are constructed to explore the relationships between the generators. The optimal system of solutions is then determined which leads the reduction of the partial differential equation into a simple ordinary differential equation. To find explicit solutions, the extended hyperbolic function method is applied. This yields various soliton solutions, including dark solitons, bright solitons, singular solitons, and periodic-singular solitons. The physical behavior of these solitons is further illustrated through 3D and 2D plots, which help visualize how the parameter constraints affect the solutions. The model is also subjected to a qualitative analysis, exploring bifurcation and chaotic behavior. Phase profiles are constructed using different values of parameters to show how the system can exhibit quasi-periodic and chaotic dynamics, when an external periodic force is applied. Different tools are used to detect chaos such as 2d phase plots and time series plots to offer a deeper understanding of the system’s dynamics and parameter variations. Hence, this study not only gives explicit soliton solutions but also reveals the complex dynamics of the reduced Ostrovsky equation. We have now included the sensitivity analysis, which examines the influence of key parameters on the system’s dynamics, enhancing the understanding of the model’s chaotic behavior.

The (3 + 1)-dimensional fractional modified Zakharov Kuznetsov (mZK) equation is one of the nonlinear models to indicate the impact of magnetic fields on weak ion-acoustic waves in plasma; made up of cool and hot electrons. The primary goal of the present study is to use the (m+G′G)-expansion technique to seek the solutions of mZK equation. The solutions are gained in the form of kink, dark, singular periodic and W-type soliton solutions. The influence of the fractional parameter on waveforms has also been examined by representing 2D and 3D graphs for distinct values of fractional-order β. Moreover, we utilize Hamiltonian system properties to confirm the stability of the solution. The (m+G′G)-expansion technique can also be used to examine the nonlinear evolution models being developed in various scientific and technological fields, such as mathematical physics and plasma physics. The soliton solutions attained by using the above technique have not been derived yet.

Until recently, scientists thought that waste was cleared from the central nervous system primarily by diffusion, with waste slowly moving from the brain tissue toward the blood vessels. However, scientists have discovered a dedicated macroscopic waste clearance system, called glymphatic system, that performs efficient waste elimination through a unique system of perivascular channels formed by astrocytes, a type of glia. To better understand the complex dynamics of fluid movements in the glymphatic system, we implemented a four-compartmental poroelasticity model of the cerebral environment and used the model in a systematic and efficient parametric study aided by machine learning, a physiologically inspired perceptron method, to explore the functional impact of water transfer coefficients. The results suggested that the model captured the transport phenomenon of human brain. Moreover, within a specific distribution range of water transfer coefficients, the model indirectly predicts the existence of the glymphatic system, providing theoretical support to the glymphatic theory. Our study also demonstrates the feasibility of discovering physiological properties of complex biological systems through computer-aided modeling. Meanwhile, the presented novel method shown its potential for parametric study.

In this work, we analysis the novel behavior of solitons to the nonlinear evolution equations describing the ionic currents microtubule and Mikhaillov-Novikov-Wang dynamical equations under an auxiliary equation approach. As a result, a variety of solitons solutions are achieved such as singular bright solitons, kink solitons, singular dark solitons and anti-kink solitons. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering including nonlinear optics, material energy, soliton wave theory, computational fluid mechanics, system identification, earthquake modeling, water wave mechanics, signal transmission, and optical fibers. We designed the utilized approach to be reliable and accurate, with precise for analytical results.