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The damped Korteweg-de Vries (D-KdV) equation is a significant extension of the Korteweg–de Vries equation, which plays an important role in understanding the complex dissipative wave system. This study explores the historical evolution and mathematical properties of the damped KdV equation. The auxiliary equation approach is utilized on nonlinear damped KdV equation to extract the newly exact solitary wave results. New solutions are extracted in bright solitons, kink wave solitons, mixed dark-bright solitons, anti-kink wave solitons, periodic solitons, dark solitons, and solitary wave structures. The physical behavior of secured solutions are visualizing in contour, 3-D, and 2-D plots based on numerical simulation with the computational software Mathematica. The proposed approach is employed, offering a powerful mathematical tool for analyzing the effects of damping on wave dynamics. Through this investigation, the study sheds light on the captivating interplay between physics and mathematics in the realm of wave phenomena. These newly constructed solutions are important in nonlinear acoustics, optical fiber, nonlinear optics, soliton wave theory, plasma physics, and ion acoustic wave phenomena. As a result, our proposed method is directed, concise, and practical for various kinds of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this research work, the nonlinear Landau–Ginzburg–Higgs model under investigation on the base of auxiliary equation method. The nonlinear Landau–Ginzburg–Higgs equation mainly describe nonlinear wave propagation, categorizes wave velocity, and materializes several phenomena through the dispersive system. The secured optical soliton solutions are interested, more general and some novel in kink solitons, bright solitons, periodic singular solitons, anti-kink solitons and dark solitons. The physical structure of some explored solitons solutions are represented graphically as contour, 2D and 3D plots by utilizing the symbolic computational tool Mathematica. The secured soliton will be play important role to the investigation of nonlinear phenomena in the various domains of nonlinear sciences and engineering such as optical fibers, nonlinear dynamics, communication system, solitons wave theory, nonlinear optics, transmission system, electronic engineering and ocean engineering. The auxiliary equation method is a basic analytical, powerful, efficient approach for the investigation of optical soliton inside the nonlinear Landau–Ginzburg–Higgs equation and also other nonlinear equations. [ABSTRACT FROM AUTHOR]

This work tackles the Heisenberg ferromagnet-type integrable Akbota equation. The Akbota equation is significant model to visualize and study the surface geometry and curve analysis. The Akbota equation is an integrable coupled system of differential equations with soliton solutions. It is a crucial tool for researching nonlinear phenomena in differential geometry of curves and surfaces, magnetism, and optics. The generalized projective Riccati equation method, the Sardar sub-equation method, and the G ′ G 2 -expansion method are the three separate analytical techniques used in this work. By using these approaches, exact analytical solutions for soliton waves are obtained, including dark, bright, singular, singular periodic, trigonometric, and hyperbolic waves. The creation of theoretical frameworks and the generalization of findings are made possible by analytical solutions. Researchers can frequently find patterns and relationships that apply more broadly by developing analytical solutions to particular cases, which results in the development of new theories and principles. The manuscript includes graphical representations, such as contour plots and two- or three-dimensional visualizations, in addition to theoretical derivations. These examples examine the propagation properties of the obtained soliton solutions and provide a promising basis for further research. Before this study, there is not existing any study in which, someone used these approaches and found solitons solutions. [ABSTRACT FROM AUTHOR]

In this research work, we extracted the variety of newly optical soliton solutions which describe the wave propagation in nonlinear low-pass electrical transmission lines model by utilizing the auxiliary equation method. The secured solitons solutions yield a variety of typical soliton shapes, including dark solitons, periodic singular optical solitons, combined bright and dark solitons, kink wave solitons, bright solitons, ant-kink wave solitons, and solitary waves. The physical structure of extracted soliton solutions visualized in three different graphically structures such as three-dimension, two-dimensional and contour plotting on the choices of some constant parameters by utilizing the numerical simulations. This study explored optical solitons, solitary wave solutions, exact solitons for improving the performance of nonlinear low-pass electrical transmission systems. It provides an overview of solitons, their relevance, and stability principles. It also presents the mathematical formulation of the nonlinear low-pass electrical transmission lines model and discusses its implications for signal propagation. The secured soliton solutions have many applications in engineering and science such as nonlinear optics, fiber optics, laser optics, nonlinear dynamics, ocean engineering, electronic engineering, electrical engineering, computing engineering, power engineering and several other different kinds of physical sciences. The whole study shows that the suggested method is more powerful, effective, simple, and strong for looking into different types of nonlinear models involve in nonlinear sciences and the engineering presentation field. [ABSTRACT FROM AUTHOR]

In this article, the main objective is to analytical investigation of the third order Klein-Fock-Gordon eation and the nonlinear Maccari’s system. The Klein-Fock-Gordon equation have vital applications in quantum field theory, article physics, condensed matter physics, astrophysics and cosmology. On the other hand, the nonlinear Maccari’s system significantly explain the neural dynamics, cardiac rhythms, population dynamics and case study for theoretical analysis and the development of mathematical techniques. In order to develop the analytical exact soliton solutions for these considered nonlinear models, the modified Kudryashov’s and extended Kudryashov’s methods are utilized and numerous kinds of soliton wave structures constructed such as dark soliton, bright soliton, dark-bright soliton and exponential solutions which are not discussed before this study along with utilized analytical techniques. The obtained soliton solutions describe the propagation of spin-0 particles like mesons behave according to a relativistic wave equation in quantum field theory. The constructed soliton wave structures of the Klein-Gordon equation represent localized, stable, and particle-like excitations of the scalar field described by the equation and can be interpreted as "quasi-particles" or "wave packets" which propagate through the field while maintaining their shape and energy. The nonlinear Maccari’s system’s soliton profiles offer potential solutions for issues like information processing, signal transmission, and pulse shaping. They also provide a framework for comprehending and modifying wave-like phenomena in complex systems. The graphical demonstration of their propagation in three-dimensional, contour and two dimensional is presented with suitable parametric values. [ABSTRACT FROM AUTHOR]

The optical soliton is a hot topic in studying. In the current research, we utilize the modified extended direct algebraic method to derive optical soliton solutions for the non-linear Kairat-X (K-X) equation, which considers second-order spatiotemporal dispersion and the group velocity dispersion. The significant Kairat-X equation is considered to develop the soliton solutions because, Solitons are valuable because they are stable, retain their shape during propagation, and have applications ranging from telecommunications and signal processing to quantum physics and materials science. This equation describe how optical solitons propagate through non-linear media. Prior to this study, there was no existing research in which such solutions could be found. Our research yields various types of soliton solutions, including bright, dark, and singular solitons, as well as periodic and exponential solutions. Under specific parameters, we generate 2D, 3D, and contour graphs to visualize the propagation of optical soliton, allowing us to showcase its precise representation of physical phenomena. This work contributes to the advancement of our understanding of optical solitons and their behavior in non-linear optical systems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the integrable motion of induced curves using the Kuralay equation, which is a complex integrable coupled system. The soliton solutions derived from Kuralay equation are supposed to represent the most advanced research in several significant phenomena, including optical fibers, nonlinear optics, and ferromagnetic materials. Analytical methods are used to obtain traveling wave solutions for this model as the Cauchy problem cannot be addressed by the inverse scattering transform. In order to find the solitary wave solutions, the new extended direct algebraic and Nucci’s reduction approaches are taken over. As a result, the new extended direct algebraic method provides singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane, and combined bright-dark soliton solutions. The Nucci’s reduction technique develops the first integral of differential equation to discuss the conservation and exact solutions. To ensure the sensitivity of the study, the effect of waves on the propagation of solitons and the sensitivity of the model is examined. To illustrate how the fitting values of the system parameters may be utilized to anticipate the behavioral reactions to pulse propagation, the resulting solutions are visually shown in 2D and 3D charts. [ABSTRACT FROM AUTHOR]

This paper investigates the Shynaray-IIA equation (S-IIAE), a type of partial differential parametric equation, using the Φ 6 -model expansion method. The Shynaray-IIA equation is coupled system of differential equations which is integrable and contain soliton solutions. It addresses the integrable motion of space curves and have a lot of applications in nonlinear optics, water waves, plasma physics and other modern sciences. Prior to this study, no previous research has achieved solutions of this kind. In order to fill this gap, the Φ 6 -model expansion method is employing to Shynaray-IIA equation and obtains new exact solutions for the perturbed form of the Shynaray-IIA equation, including bright, singular, periodic, and combined singular soliton solutions. These findings enhance our understanding of the equation's nonlinear dynamical properties and offer a practical and efficient approach to solving various nonlinear partial differential equations. The research also explores specific parameter values that meet constraint conditions to reveal the dynamic behavior of the solutions. To present these findings effectively, the study utilizes visually appealing graphs that highlight the solutions' characteristics. Overall, this work contributes to advancing our knowledge of the Shynaray-IIA equation and demonstrates the applicability of the Φ 6 -model expansion method in studying nonlinear systems. [ABSTRACT FROM AUTHOR]

This work aims to enhance our comprehension of the dynamical features of the nonlinear Landau-Ginzburg-Higgs evolution equation, which provides a theoretical framework for identifying various phenomena, such as the formation of superconducting states and the spontaneous breakdown of symmetries. When symmetry breaking is involved in phase transitions in particle physics or condensed matter systems, the Landau-Ginzburg-Higgs model combines the ideas of the Landau-Ginzburg theory and the Higgs mechanism. The equation plays a crucial role in characterizing the Higgs field and its related particles, including Higgs boson. In a standard model of the particle physics, Higgs mechanism explains precisely how mass is acquired. The Lie invariance requirements are taken into account by the symmetry generators. The method produces a 3 - dimensional Lie algebra of the Landau-Ginzburg-Higgs model with translational symmetry (dilation or scaling) and translations in the space and the time associated with the mass and energy conservation. It is shown to be the optimal sub-algebraic system after similarity reductions are also performed. The next wave transformation method reduces the governing system to ordinary differential equations and yields a large number of exact travelling wave solutions. The Khater approach is used to solve an ordinary differential equation and investigate the closed-form analytical travelling wave solutions for the considered diffusive system. The obtained results include a singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane, and combined bright-dark soliton solution. The results of the sensitivity analysis demonstrate how vulnerable the suggested equation is to various initial conditions. The findings are visually displayed in contour, three-dimensional, and two-dimensional forms to emphasize the features of pulse propagation. [ABSTRACT FROM AUTHOR]

This study employs the new extended direct algebraic method and improved sardar sub-equation method to investigate solitary wave solutions in the Shynaray-IIA equation, which characterizes phenomena like tidal waves and tsunamis. These methods transform complex nonlinear coupled partial differential equations into manageable algebraic equations using a traveling wave transformation. Before this study, there is not exiting any research in which someone has obtained such kind of solutions. The main goal is to enhance understanding of the Shynaray-IIA equation behavior in various scenarios. By applying the new extended direct algebraic method and improved sardar sub-equation method, the study derives solitary wave solutions using trigonometric, hyperbolic, and Jacobi functions. By adjusting specific parameters, diverse solutions are obtained, including periodic, bell-shaped, anti-bell-shaped, M-shaped, and W-shaped solitons, each pair exhibiting mathematical symmetry. The analytical soliton solutions are further visualized in both 2D and 3D representations using Mathematica 12.3, aiding in the interpretation of these complex wave phenomena. [ABSTRACT FROM AUTHOR]

This research examines the fractional Peyrard–Bishop DNA dynamical governing system, which displays the proliferation of optical pulses in field of plasma and the optical fibre. The analytical method is utilized to find travelling wave solutions because the inverse scattering transform cannot solve the Cauchy problem for this equation. The Φ 6 -model expansion method is an efficient and dependable technique for generating the generalised solitonic wave profiles with a wide range of soliton families. The main advantage of the offered analytical strategy is that it specifies a constraint for each solution to guarantee its existence. As a result, solitonic wave structures get attributes such as the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures, among others, under existence conditions that have not been explored previously. The results are represented graphically in 2-D, 3-D, and contour glimpses to illustrate the behavioural responses to pulse propagation by inferring the fitting values of system parameters. The stabilty of the considered model is discussed and develope the stability condition. The fractional parameter is responsible for reducing singularity and continuing to increase the smoothness in wave patterns. It is easy to employ the Φ 6 -model expansion method to other complicated non-linear systems and acquire solitary waves pattern. [ABSTRACT FROM AUTHOR]

The Delta and Omicron variants' system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both variants. The proposed dynamical system has heredity and memory effects, which greatly improved our ability to perceive the disease propagation dynamics. The non-singular Atangana–Baleanu fractional operator was used to forecast the current pandemic in order to meet this challenge. The Picard recursions approach can be used to ensure that the designed fractional system has at least one solution occupying the growth condition and memory function regardless of the initial conditions. The Hyers–Ulam–Rassias stability criteria were used to carry out the stability analysis of the fractional governing system of equations, and the fixed-point theory ensured the uniqueness of the solution. Additionally, the model exhibited global asymptotically stable behaviour in some conditions. The approximate behaviour of the fatal virus was investigated using an efficient and reliable fractional numerical Adams–Bashforth approach. The outcome demonstrated that there will be a significant decline in the population of those infected with the Omicron and Delta SARS-CoV-2 variants if the vaccination rate is increased (in both the symptomatic and symptomatic stages). [ABSTRACT FROM AUTHOR]

The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields. [ABSTRACT FROM AUTHOR]

The motive of the study was to explore the nonlinear Riemann wave equation, which describes the tsunami and tidal waves in the sea and homogeneous and stationary media. This study establishes the framework for the analytical solutions to the Riemann wave equation using the new extended direct algebraic method. As a result, the soliton patterns of the Riemann wave equation have been successfully illustrated, with exact solutions offered by the plane solution, trigonometry solution, mixed hyperbolic solution, mixed periodic and periodic solutions, shock solution, mixed singular solution, mixed trigonometric solution, mixed shock single solution, complex soliton shock solution, singular solution, and shock wave solutions. Graphical visualization is provided of the results with suitable values of the involved parameters by Mathematica. It was visualized that the velocity of the soliton and the wave number controls the behavior of the soliton. We are confident that our research will assist physicists in predicting new notions in mathematical physics. [ABSTRACT FROM AUTHOR]

The weakly nonlinear wave propagation that occurs in the presence of magnetic fields, in which energy is concentrated in a narrow band of wave-numbers in a dispersive and dissipative fluid. The main objective of this paper is to analyze the (2 + 1) - dimensional elliptic nonlinear Schrodinger equation under the influence of three different fractional operators. The generalized fractional soliton solutions and propagation of magnetohydrodynamics fluid in sort of solition will be visualized. The Conformable, β and M-truncated fractional operator applied to classical evolution Schrodinger equation. In order to get the analytical closed form solution, one of the generalized approach new extended direct algebraic method is utilized. The fractional nonlinear elliptic Schrodinger equation is developed in three different fractional sense. The similarity transformation technique converted the controlling fractional system to ordinary differential equations. The fractional analytical solutions such as, plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are obtained. The graphical 2-D and 3-D representation of the results is shown to express the propagation of fluid with the magnetic field by assuming the appropriate values of the involved parameters. The graphical performance of the obtained solution at various settings of parametric values and fractional order reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields. [ABSTRACT FROM AUTHOR]

The wave propagation has the significant role in the field of coastal engineering and ocean. In the geographical fields, waves are primary source of environmental process owed to energy conveyance on floating structure. This study aims to investigate the system of cold bosonic atoms in zig-zag optics lattices. The solitonic patterns of the considered model successfully surveyed by using two integrated analytical techniques new extended direct algebraic and G ′ G 2 expansion method. The exact solutions are presented by rational, trigonometric, hyperbolic and exponential functions. On the basis of solitons, we need to show that which one is more integrated and robust scheme. These solutions will help to understood the dynamics of cold bosonic atoms in zig-zag optical lattices and the several other systems. Three dimensional as well as two dimensional comparison presented for a cold bosonic atoms model solutions which are revealed diagrammatically for appropriate parameters by using Mathematica. This study will help physicists to predict some new hypothesis and theories in the field of mathematical physics. [ABSTRACT FROM AUTHOR]

This study examines the Chen–Lee–Liu dynamical equation, which represents the propagation of optical pulses in optical fibers and plasma. A new extended direct algebraic technique and Nucci's scheme are used to find new solitary wave profiles. The method covers thirty-seven solitonic wave profiles, including approximately all soliton families, in an efficient and generic manner. New solitonic wave patterns are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions. The exact fractional solution is obtained using Nucci's reduction approach. The impact of the fractional order parameter on the solution is considered using both mathematical expressions and graphical visualization. The fractional order parameter is responsible for controlling the singularity of the solution which is graphically displayed. A sensitivity analysis was used to predict the sensitivity of equations with respect to initial conditions. To demonstrate the pulse propagation characteristics, while taking suitable values for the parameters involved, 2-D, 3-D, and contour graphics of the outcomes achieved are presented. The influence of the fractional order ζ is shown graphically. A periodic-singular wave with lower amplitude and dark-singular behaviour is inferred from the graphical behaviour of the trigonometric function solution H 1 and the rational function solution H 34 from the obtained solutions, respectively. [ABSTRACT FROM AUTHOR]

A generalized (3 + 1)-dimensional nonlinear wave is investigated, which defines many nonlinear phenomena in liquid containing gas bubbles. Basic theories of the natural phenomenons are usually described by nonlinear evolution equations, for example, nonlinear sciences, marine engineering, fluid dynamics, scientific applications, and ocean plasma physics. The new extended algebraic method is applied to solve the model under consideration. Furthermore, the nonlinear model is converted into an ordinary differential equation through the next wave transformation. A well-known analytical approach is used to obtain more general solutions of different types with the help of Mathematica. Shock, singular, mixed-complex solitary-shock, mixed-singular, mixed-shock singular, mixed trigonometric, periodic, mixed-periodic, mixed-hyperbolic solutions are obtained. As a result, it is found that the energy-carrying capacity of liquid with gas bubbles and its propagation can be increased. The stability of the considered model is ensured by the modulation instability gain spectrum generated and proposed with acceptable constant values. Two-dimensional, three-dimensional, and contour surfaces are plotted to see the physical properties of the obtained solutions. [ABSTRACT FROM AUTHOR]

This study deals with the perturbed Chen-Lee-Liu governing mode which portrays the propagating phenomena of the optical pulses in the discipline of optical fiber and plasma. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we use an analytical approach to find traveling wave solutions. One of the generalized techniques Φ 6 - model expansion method is exerted to find new solitary wave profiles. It is an effective, and reliable technique that provides generalized solitonic wave profiles including numerous types of soliton families. As a result, solitonic wave patterns attain, like Jacobi elliptic function, periodic, dark, bright, singular, dark-bright, exponential, trigonometric, and rational solitonic structures, etc. The constraint corresponding to each obtained solution provides the guarantee of the existence of the solitary wave solutions. The graphical 2-D, 3-D, and contour visualization of the obtained results is presented to express the pulse propagation behaviors by assuming the appropriate values of the involved parameters. The Φ 6 - model expansion method is simple which can be easily applied to other complex non-linear models and get solitary wave structures. [ABSTRACT FROM AUTHOR]

This study critically examines the nonlinear Lonngren wave model, a mathematical representation used to describe waves in shallow water environments such as beaches, lakes, and rivers, as well as electrical signals in semiconductor materials like tunnel diodes. The symmetry generators take into account the Lie invariance criteria. A three-dimensional Lie algebra can be obtained by using the suggested methodology. It links translation symmetries in space and time, respectively, with conservation of mass and energy. The third symmetry pertains to scaling or dilation. Recognizing the constraints of the inverse scattering transform in addressing the Cauchy problem for this equation, the investigation adopts new auxiliary equation approach to ordinary differential equations. This approach yields closed-form analytical solutions for the wave model. The obtained solutions include a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, smooth soliton, trigonometric solution, mixed singular solution, complex solitary and singular solution. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the total energy of the system in terms of its position and momentum. Chaos analysis is performed to describe periodic, quasi-periodic, and chaotic behaviours, and a sensitivity analysis highlights the model's sensitivity to different wave numbers. Graphical representations of selected solutions are presented in both 3-D and 2-D, illustrating the behaviour under specific physical conditions. The study concludes by identifying a comprehensive set of local conservation laws for the nonlinear Lonngren wave equation, applicable to arbitrary constant coefficients, using conservation laws multipliers. [ABSTRACT FROM AUTHOR]

The nonlinear Whitham-Broer-Kaup dynamical equation under investigation on the base of auxiliary equation approach. We secured the exact soliton solutions of nonlinear Whitham-Broer-Kaup dynamical equation in this investigation by computational simulation. The Whitham-Broer-Kaup equation is the important nonlinear evolution equation which describe changes in the wave envelope as it propagates through a dispersive medium. In this study, the explored exact soliton solutions have different and interested physical shape like kink solitons, bright solitons, anti-kink solitons, dark solitons, periodic wave solitons and mixed solitons. The graphical analysis of explored soliton solutions visualized in 2-dim, 3-dim and contour plotting by utilized the numerical simulation with computational software Mathematica. The complete investigation shed light to the utilized approach that more powerful, effective, concise and easy to use for the exploration of analytical soliton solutions of variety of nonlinear equations. [ABSTRACT FROM AUTHOR]

In this study, the generalized perturbed-KdV partial differential equation is examined. Furthermore, symmetry generators address the Lie invariance criteria. The suggested approach produces the Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. The optimal system of the obtained system developed. By using Lie Group methods, the generalized perturbed-KdV partial differential equation is changed using suitable similarity transformations through a system of highly nonlinear ordinary differential equations. The new extended direct algebraic approach is applied to get the soliton solutions. As a result, a plane solution, periodic stumps, compacton, smooth soliton, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, peakon soliton, anti-peaked with decay, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are developed. The behavior of certain solutions is shown in 3-D and 2-D for specific values of the physical components in the studied equation. The outcomes hold significance for elevating research to a more impactful and effective level. The whole set of local conservation laws for the generalized perturbed-KdV equation for any arbitrary constant coefficients is found by applying the conservation laws multiplier. These findings are pivotal for advancing the current understanding and pushing the boundaries of knowledge to new heights. [ABSTRACT FROM AUTHOR]

The present study examines the nonlinear time-fractional model in the sense of a solitonic structure. A non-linear Schrödinger equation has applications in light scattering, indirect optical pulses as well as planer waves and to Bose-Einstein condensates enclosed in an anisotropic-shaped cigar, in a mean-field state, etc. A new extended direct algebraic method is utilized to get the soliton solutions with modified M-truncated and Atangana–Baleanu fractional operators which have Mittag-Leffler kernel. The obtained solutions contain new families of functions such as trigonometric, hyperbolic, rational, and exponential functions. The graphical 2D, 3D, contour, and also 3D spherical presentation pictorial the analysis with the feasible parametric values. On the evidence of the acquired solutions, it can be presumed that this technique is more effective and generalized to obtain solutions of many other non-linear partial differential equations that appear in different scientific disciplines. [ABSTRACT FROM AUTHOR]

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures. [ABSTRACT FROM AUTHOR]

The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1 + 1) − dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci's scheme and a new extended direct algebraic method. The new extended direct algebraic approach provides an easy and general mechanism for covering 37 solitonic wave solutions, which roughly corresponds to all soliton families, and Nucci's direct reduction method is used to develop the first integral and the exact solution of partial differential equations. Thus, there are several new solitonic wave patterns that are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, a mixed trigonometric solution, a trigonometric solution, a shock solution, a mixed shock singular solution, a mixed singular solution, a complex solitary shock solution, a singular solution, and shock wave solutions. The first integral of the considered model and the exact solution are obtained by utilizing Nucci's scheme. We present 2-D, 3-D, and contour graphics of the results obtained to illustrate the pulse propagation characteristics while taking suitable values for the parameters involved, and we observed the influence of parameters on solitary waves. It is noticed that the wave number α and the soliton speed μ are responsible for controlling the amplitude and periodicity of the propagating wave solution. [ABSTRACT FROM AUTHOR]

The new extended direct algebraic technique is used to attain the new exact solitary wave solutions of the nonlinear fractional couple Drinfeld–Sokolov–Wilson system (DSWS). This DSWS has a lot of applications in water waves, fluid mechanics, quantum mechanics, chemistry, and biology. This system is developed in the perspective of fractional theory of calculus, for this purpose Riemann–Liouville fractional derivative (RLFD), β -fractional differential operator and Atangana–Baleanu fractional derivative (ABFD) are utilized. The attained solutions contain some new families such as trigonometric functions, hyperbolic functions, rational functions, logarithmic functions, and exponential functions. The graphically analysis and comparison of solutions are plotted by taking appropriate values of the constant parameters to enhance the understanding of physical real-life phenomena. The approach is said to be compatible and effective as a scientific approach for the examination of the various spacetime fractional models in engineering and physics concerning real-life issues. [ABSTRACT FROM AUTHOR]

The (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation is a significant model that describes the behaviour of conserved scalar nucleons and their connection to neutral scalar masons. This system exhibits self-formation and depends on various free parameters. It is commonly used to study the dynamics of scalar nucleons and neutral scalar masons. This research article used an extended direct algebraic technique to obtain exact travelling wave solutions for the Nizhnik–Novikov–Veselov system. The obtained soliton solutions include various types, such as multi-periodic, periodic, plane, singular, bright, dark, and flat kink-type wave solitons. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations in 3D, contour, and 2D surfaces allow us to visualise the behaviour of the solutions as the parameters change and the physical interpretation for these solutions is obtained. Additionally, we conduct a sensitivity analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form travelling wave solitary solutions for a wide range of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this work, we prove Ostrowski–Mercer inequalities for differentiable harmonically convex functions. It is also shown that the newly proved inequalities can be converted into some existing inequalities. Furthermore, it is provided that how the newly discovered inequalities can be applied to special means of real numbers. [ABSTRACT FROM AUTHOR]

This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ 6 -model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature. [ABSTRACT FROM AUTHOR]