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This study focuses on the Fokas system equation, which explains the nonlinear pulse propagation in single-mode fibres. The proposed new sub-equation approach and the modified Khater's method extract a more efficient and broad range of soliton solutions. These include multiple anti-kink, anti-kink, flat anti-kink, combined bright-periodic, multiple bright, v-shaped, bell-shaped, and flat bell-shaped singular soliton. The primary innovation of this paper lies in its discovery of a set of novel soliton solutions that provide insight into the interaction mechanisms of nonlinear waves within the realm of nonlinear physics. The travelling wave patterns of the model are graphically presented with suitable parameter values using the modern software Mathematica. The visual representation of the solutions in 3D, 2D, and contour surfaces enhances understanding of parameter impact. We conducted a sensitivity analysis on a newly developed dynamic framed structure that controls pulses using the velocity factor of the soliton wave. The proposed model's dynamics were observed and presented as bifurcation analysis, quasi-periodic chaotic, periodic systems and quasi-periodic. We anticipate that our research findings will catalyze advancing the study of nonlinear theory and its applications in optical fibres. This analysis confirms the effectiveness and reliability of the method employed, demonstrating its applicability in discovering travelling wave solitons for a wide range of nonlinear evolution 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]

In this article, we present an idea to attain the variety of novel optical solitons solutions for nonlinear time-fractional Clannish Random Walker's parabolic (CRWP) equation in the manner of β -derivative using a well-known analytical technique namely, extended direct algebraic method. A diversity of bright, singular, dark, periodic-singular and combo dark-bright solitons solutions are assembled in the shape of rational, hyperbolic and exponential functions. Some of the extracted results are sketched in the pattern of 3D, 2D and contour plots, for the purpose to demonstrate the physical behavior of the attained solutions. We also investigate the influence of the fractional order parameter on the obtained solutions which demonstrates how changes in this parameter affect the properties and behavior of the soliton solutions in the context of the time-fractional CRWP model. The novelty of the extracted solutions is determined by comparing with some other solutions which are already listed in the works for the CRWP equation, which shows the effectiveness and authenticity of the proposed method. The under consideration method is also utilized to any other nonlinear integer and fractional model appears in physics and engineering. [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]

By applying the Sardar subequation method, we study new coupled Konno–Oono (CKO) equation and obtained soliton type solutions in the form of bright, dark, singular, periodic singular and combined solutions. In nonlinear sciences, CKO equation is very important and has applications in magnetic field. In order to better explanation of the developed solutions, different sketches of some solutions are drawn. The proposed approach is simple, direct and effective to obtain new results for nonlinear equations in mathematical sciences. [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]

Two efficient integration schemes, new extended hyperbolic function and generalized tanh are employed to discover optical soliton solutions to magneto-optic waveguides that retains anti-cubic form of nonlinear refractive index. Bright, dark, periodic singular, singular, and combo soliton solutions have created. These solutions expose the comprehensive variety of soliton solutions. [ABSTRACT FROM AUTHOR]