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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]

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]

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]

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 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 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 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 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]