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This paper presents a comprehensive analysis of the (1 + 1) -dimensional Klein-Gordan equation which plays a significant role in various areas of theoretical and applied physics. The main focus of this research centers on several key areas. First, infinitesimal generators of symmetries were found by Lie symmetry invariance analysis. Then, using the adjoint representation, an ideal system was created based on the found Lie vectors. Secondly, by utilizing the analytical approach, namely the modified Sardar sub equation method, we systematically derive various novel soliton solution in the form of dark, bright, periodic, singular, combo, hyperbolic as well as mixed trigonometric. Finally, bifurcation analysis is performed at the system's fixed points, revealing chaotic behavior when an external force is introduced into the dynamic system. To identify the chaotic characteristics, a range of tools, such as 3D and 2D phase plots, time series, Lyapunov exponents, and multistability analysis, are utilized. Additionally, the sensitivity analysis of the model is examined under different initial conditions. These results enhance the understanding of nonlinear wave phenomena in mathematical physics and hold potential applications across numerous scientific disciplines. [ABSTRACT FROM AUTHOR]

Based on a modified Snyder–Mitchell model in nonlocal photorefractive media, spatiotemporal optical solitonary solutions are obtained and the formation and transmission characteristics of spatial solitons under the control of 2D optical lattice are studied by using self-similarity algorithm. The simulations show that the lattice order, modulation intensity and beam width significantly affect the shape and stability of spatial soliton in photonic lattice. The results have important guiding significance for the experimental study of spatial solitons. [ABSTRACT FROM AUTHOR]

This investigation discusses the modified M-truncated form of the perturbed Chen–Lee–Liu (pCLL) dynamical equation. The pCLL equation is a generalization of the original CLL equation, which describes the propagation of optical solitons in optical fibers. The pCLL equation includes additional terms that account for various influences such as chromatic dispersion, nonlinear dispersion, inter-modal dispersion, and self-steepening. A new version of the generalized exponential rational function method is utilized to obtain multifarious types of soliton solutions. Moreover, the planar dynamical system of the concerned equation is created using a Hamiltonian transformation, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to better understand the dynamic behavior of these solutions. The study's findings could be extremely useful in solving additional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Highly important is a three-dimensional nonlinear partial differential equation because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that describes how the system changes in time. Thus, this article comprehensively reveals the investigation carried out on a (3+1)-dimensional generalized fifth-order Zakharov–Kuznetsov equation with power-law as well as dual power-law nonlinearities analytically, where the fifth-order term involved is regarded as a dispersion perturbation term. We utilize the well-celebrated Noether's theorem to comprehensively construct conserved currents of the underlying equation. A detailed Lie group analysis of the understudied model consisting of power-law nonlinearities is further performed. This involves performing reductions of the underlying models using their Lie point symmetries. In consequence, various invariants are found. In addition, the equation reduces to diverse ordinary differential equations using its point symmetries and consequently diverse solutions of interest were achieved. Moreover, we derive some solitary wave solutions by invoking the newly introduced logistic function technique for some particular cases of the equation under consideration. In consequence, we achieve some exponential function solutions. In addition, the physical meaning of the results is put on the front burner by revealing the wave dynamics of these solutions via graphical depictions. Finally, the significance of the robust and detailed findings in the work are further corroborated with various real-world applications. [ABSTRACT FROM AUTHOR]

This work simulates the (1+1)-dimensional nonlinear perturbed Schrödinger model (NLPSM ). Hydrodynamics, elastic media, nonlinear optical fiber communication, and plasma physics are just a few of this model's mathematical physics and engineering applications. The study aims to accomplish two main objectives. First, it seeks to find unique soliton solutions such as solitary, dark, periodic, and plane wave solutions that haven't been found in the literature before using the modified Sardar sub-equation approach (MSSEA ). Second, a novel approach to analysis called bifurcation analysis is used to investigate the dynamic behavior of the model. Physical compatibility findings are supported by density, 3-D, and 2-D illustrations made with parametric variables. The analysis shows that the approach used to quickly acquire complete and typical answers was successful. This approach works well for solving challenging problems in physics, engineering, mathematics and fiber optic phenomena. [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 study, we studied the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) system using the improved Ricatti equation method. cmKdV are nonlinear and coupled partial differential equations that arise in various fields of applied science and engineering, such as ferromagnetic materials and optical fibers. When the method is applied to cmKdV, we successfully derive exact soliton solutions that accurately describe the wave propagation behavior of the system under consideration. The obtained results include trigonometric and hyperbolic function solutions. The results obtained are concise and offer a deeper insight into the dynamics and characteristics of cmKdV. Traveling wave solitons are plotted in 2D and 3D to demonstrate the wave propagation phenomena in the cmKdV model, which are in the form of kink, bright, dark, singular solitons, and periodic solitary wave structures. The method recovers many solutions compared with the existing methods in the literature, indicating that the proposed method is a powerful and valuable approach for achieving analytical solutions to a wide range of nonlinear partial differential equations [ABSTRACT FROM AUTHOR]

In this paper, we investigate two types of nonlinear Schrödinger equations (NLSE): the unstable NLSE and the modify unstable NLSE. These equations describe the time evolution of disturbances in unstable media. To solve the proposed equations, we employ the variational principle method that involves selecting trial functions based on the Jost function in different forms. Also, these ansatz functions should be continuous at all intervals and may contain single or two nontrivial variational parameters. After that, we use these trial functions to find the functional integral and Lagrangian of the system without any loss. Besides, we use the amplitude ansatz method to explore new soliton solutions. The obtained results include various solitons, such as bright soliton, dark soliton, bright–dark solitary wave solutions, rational dark-bright soliton solutions, and periodic solitary wave solutions. The results will be displayed through different types of graphs, including 2D, 3D, and contour plots, which effectively highlight their outcomes. These solutions have essential applications in the fields of applied science and engineering. Also, they are stable and analytical solutions. The offered techniques can be utilized to solve numerous nonlinear models in mathematical physics and various applied sciences fields. [ABSTRACT FROM AUTHOR]

The analytical study of Boiti-Leon-Manna-Pempinelli (BLMP) equation is presented in this research paper. In this study, two exact methods are utilized to attain the exact solution of proposed equation. The generalized projective Riccati equations method and modified auxiliary equation method are simple and effective techniques, which have been used to attain the exact soliton solutions of BLMP equation. Some novel exact solution of BLMP equation are acquired using of proposed methods. The obtained solutions contain rational, geometric, hyperbolic functions. The graphical simulations of attained solutions are represented by plotted graphs. The plotted graphs show different solitons patterns such as kink solitons, anti-kink soliton, dark singular soliton, bright singular soliton, dark-bright singular solition and some other singular solitons. Mathematical modeling, analysis of physical phenomena and dynamical processes can yield solutions that enhance our understanding of their dynamics, which can be leveraged to gain valuable insights into the behavior and characteristics of these systems. [ABSTRACT FROM AUTHOR]

The primary objective of this study is to extract nonlinear wave patterns from the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli (4D-BLMP) equation, considering both constant and time-dependent coefficients, which is used widely to describe the incompressible fluid. By employing the amended extended tanh-function method, we successfully obtained innovative solutions in the form of combo dark bright, hyperbolic or lumps, periodic, and singular mix solitons solutions, and others. To ensure the utmost precision and reliability of our findings, we rigorously confirm them using the robust Mathematica software. These solutions hold paramount importance in the domains of in the study of incompressible fluids and acoustic waves, enriching our understanding of the foundational physical principles embedded within the equation. The study visually presents the computed wave solutions using 2D, 3D, and contour plots, effectively representing the internal structure of the phenomenon. This study proves that the computational method used is efficient, brief, and widely applicable, making it valuable to engineers who work with engineering models and dynamical models. This research can help to better understand physical phenomena in many areas of applied physics, particularly in the study of incompressible fluids and acoustic waves. [ABSTRACT FROM AUTHOR]

The nonlinear space–time fractional Sasa–Satsuma and Schrödinger–Hirota equations with beta derivative describe optical soliton, photonics, plasmas, neutral scalar masons, and long-surface gravitational waves in the real world. Through the fractional wave transform, the models are converted into a single wave variable equation. In this article, we examine a range of compatible, useful, and typical wave solutions expressed in the forms of hyperbolic, trigonometric, and rational functions uniformly through the ( Q ′ / Q , 1 / Q )-expansion approach. When specific parameter values are set, the generalized wave solutions exhibit a wide range of shapes, including asymptotic, anti-asymptotic, dark-optical, breather, lump-periodic, kink, kink-bell-shaped, homoclinic-breather, bright, dark, and periodic solitons that resemble periodic breathing patterns. We also investigate the effect of the fractional parameter δ into the wave profile, revealing a clear correlation between changes in the fractional order derivative δ and variation in the soliton's shape. The results underscore the use of this approach for the exploration of diverse nonlinear fractional systems within the context of beta derivatives. Varying the fractional-order δ and maintaining specific fixed parameter values, we depict 3D-surface, 2D-surface, density, and contour plots to visualize some of the derived solutions. [ABSTRACT FROM AUTHOR]

This paper investigates a novel method for exploring soliton behavior in ion transport across biological membranes. This study uses the Hirota bilinear transformation technique together with the Poisson–Nernst–Planck equation. A thorough grasp of ion transport dynamics is crucial in many different scientific fields since biological membranes are important in controlling the movement of ions within cells. By extending the standard equation, the suggested methodology offers a more thorough framework for examining ion transport processes. We examine a variety of ion-acoustic wave structures using the Hirota bilinear transformation technique. The different forms of solitons are obtained including breather waves, lump waves, mixed-type waves, periodic cross-kink waves, M-shaped rational waves, M-shaped rational wave solutions with one kink, and M-shaped rational waves with two kinks. It is evident from these numerous wave shapes that ion transport inside biological membranes is highly relevant, and they provide important insights that may have an impact on various scientific disciplines, medication development, and other areas. This extensive approach helps scholars dig deeper into the complexity of ion transport, illuminating the complicated mechanisms driving this essential biological function. Additionally, to show the physical interpretations of these solutions we construct the 3D and their corresponding contour plots by choosing the different values of constants. So, these solutions give us the better physical behaviors. [ABSTRACT FROM AUTHOR]

We have studied the dynamics of various kinds of optical dark solitons like, chirped, chirp-free, M-shaped & wing shaped dark solitons using higher-order nonlinear Schrödinger equation. To obtain the exact analytical solution, we employed mathematical techniques such as the extended rational sinh-cosh and sin-cos methods. Our investigation shows that one can manipulate the shape of both chirp and chirp free dark solitons by properly tuning the magnitude of the self steepening and self phase modulation. The stability of the obtained dark soliton solutions are verified by using linear stability analysis. [ABSTRACT FROM AUTHOR]

This manuscript delves into the examination of the stochastic fractional derivative of Drinfel'd-Sokolov-Wilson equation, a mathematical model applicable in the fields of electromagnetism and fluid mechanics. In our study, the proposed equation is through examined through various viewpoints, encompassing soliton dynamics, bifurcation analysis, chaotic behaviors, and sensitivity analysis. A few dark and bright shaped soliton solutions, including the unperturbed term, are also examined, and the various 2D and 3D solitonic structures are computed using the Tanh-method. It is found that a saddle point bifurcation causes the transition from periodic behavior to quasi-periodic behavior in a sensitive area. Further analysis reveals favorable conditions for the multidimensional bifurcation of dynamic behavioral solutions. Different types of wave solutions are identified in certain solutions by entering numerous values for the parameters, demonstrating the effectiveness and precision of Tanh-methods. A planar dynamical system is then created using the Galilean transformation, with the actual model serving as a starting point. It is observed that a few physical criteria in the discussed equation exhibit more multi-stable properties, as many multi-stability structures are employed by some individuals. Moreover, sensitivity behavior is employed to examine perturbed dynamical systems across diverse initial conditions. The techniques and findings presented in this paper can be extended to investigate a broader spectrum of nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

In this work, the (2+1)-dimensional Painlevé integrable Burgers equation is investigated. By applying a certain unified method, some analytical solutions, involving rational functions, trigonometric functions and hyperbolic functions, are achieved. In order to predict the wave dynamics, several three-dimensional and two-dimensional graphs and contour profiles are constructed. Bright, dark, periodic, kink, anti-kink, singular, singular periodic, bell-shaped waves are thus obtained. The dynamics of these solutions can be illustrated graphically by choosing appropriate values for the parameters involved. Due to the presence of arbitrary constants in these derived solutions, they can be used to explain a variety of qualitative traits present in wave phenomena. The approach is efficient to algebraic computation and it can be used to categorize a wide range of wave forms, as shown by the demonstrated soliton solutions. Travelling wave solutions are converted into solitary wave solutions when certain values are set for the parameters. Using the Wolfram program Mathematica, we sketch the figures for various values of the associated parameters in order to closely examine the obtained solitons. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE), which governs the propagation of pulses in optical fiber while including the effects of second, third, and fourth-order dispersion, is crucial for a comprehensive understanding of pulse propagation in optical communication systems. It assists engineers and scientists in optimizing and controlling the behavior of ultra-short pulses in complex and real-world optical systems. In this study, we solve the generalized NLSE for the pulse envelope E (z , τ) with ν -time derivative by employing the Sardar subequation method (SSM). We obtain new soliton solutions corresponding to the relevant parameters of this technique. Additionally, conditions depending on the parameters of optical pulse envelope E (z , τ) are provided for the existence of such soliton structures. Furthermore, the solitary wave solutions are expressed in the form of generalized trigonometric and hyperbolic functions. The dynamic behaviours of the solutions are revealed with specific values of the parameters that satisfy their respective existence criteria. The results indicate that SSM demonstrates high reliability, simplicity, and adaptability for use with various nonlinear equations. [ABSTRACT FROM AUTHOR]

In this study, we examine three essential types of nonlinear Schrödinger equation (NLSE), which are NLSE with high nonlinearity, resonant NLSE with parabolic law nonlinearity, and fourth-order dispersive NLSE. We use two powerful and easily understandable techniques, namely the variational principle method and amplitude ansatz method, to investigate these equations. We derive the functional integral and the Lagrangian of these equations to describe the system's dynamics. Additionally, we utilize different forms of trial ansatz functions based on the Jost function with one and two nontrivial variational parameters to substitute them in the Lagrangian and determine the amount of eigenvalue. We also obtain new solitary wave solutions of the proposed equations, which include bright soliton, dark soliton, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. The solutions accepted from these techniques have a broad scope of applications in various areas of physics and other applied sciences. The results will be illustrated in graphical representations such as 2D, 3D, and contour plots, which underline their effectiveness. These techniques can be employed to solve numerous other nonlinear models that appear in mathematical physics and various other applied sciences fields. [ABSTRACT FROM AUTHOR]

The nonlinear evolution equations (NLEEs) play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Drinfeld–Sokolov–Wilson (DSW) equation and the Burgers equation are the fundamental equations occurring in various areas of physics and applied mathematics, such as nonlinear acoustics and fluid mechanics. The new generalized (G ′ / G) -expansion method is an effective and more powerful mathematical tool for solving NLEEs arising in applied mathematics and mathematical physics. In this article, we investigate further exact solutions as well as soliton solutions to these couple of nonlinear evolution equations by executing the new generalized (G ′ / G) -expansion method. A large number of soliton solutions, including single soliton, bell-shaped soliton, kink-shaped soliton, singular kink soliton, singular soliton, periodic soliton, irregular periodic soliton solutions, and others, have been retrieved. Each of the derived solutions includes an explicit function of the variables in the equations under consideration. We provide some 3D plots to visualize and realize the characteristics of these solutions. It has been established that the suggested techniques are more potential and successful at obtaining soliton solutions for nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this study, we explore a captivating (3+1)-dimensional negative-order Korteweg–de Vries Calogero–Bogoyavlenskii–Schiff equation, which combines elements of the Korteweg–de Vries equation and the Calogero–Bogoyavlenskii–Schiff equation. Our research investigates how this model characterises long-wave interactions and its relevance in mathematics, physics, and engineering. We employ unified and singular manifold methods to obtain precise travelling wave solutions expressed in various functional forms. By using Maple and Mathematica software to extract valid solutions, including kink-like soliton, singular periodic wave solution, anti-kink solutions, and singular solitons. These methodologies have shown impressive efficiency in solving complex nonlinear equations, offering precise solutions, and streamlining mathematical processes through transformations. This leads to quicker and more accurate outcomes in diverse scientific and engineering applications. Our findings underscore the model's superiority over existing methods and its importance in comprehending applied mathematical processes, as demonstrated through 3-D and 2-D graphical representations. [ABSTRACT FROM AUTHOR]

This study addresses the integrable Akbota equation which is a Heisenberg ferromagnet-type equation and has a significant study of the curve and surface geometry. The conservation laws of the model are computed using the multipliers approach. The Jacobi elliptic function solutions for the model are derived by utilizing the new extended auxiliary equation method. The Jacobi elliptic functions solutions are degenerated to dark, bright, singular and singular periodic solitons in their limit conditions. Further, the modulation instability analysis of the equation is studied. Physical interpretations of the obtained results are represented graphically. To the best of our knowledge, the obtained results in this article are novel and have not been reported before. [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 research, we utilized the auxiliary equation technique to invent the soliton results of the nonlinear third-order Klein–Fock-Gordon (KFG) equation. With the capability of explicit approach soliton results has been secured on the base of computational software. As a result, various solitary wave solutions are produced and shown in hyperbolic functions. The procedure delivers more general and wide-ranging soliton solutions separated with parameters and, for different values of constant parameters, reveals different shapes of solitons form, like dark soliton, bright soliton, combined bright and dark solitons and other types of solitons. It has been established that the method used to examine the nonlinear model is reliable, companionable, and reasonably good. Also, the method shows that it can be utilized on other types of nonlinear equations in a thorough way. [ABSTRACT FROM AUTHOR]

The investigation revolved around the study of the time fractional Gardner equation, which was examined in terms of the conformable derivative. The reduction of the Gardner equation to an integer order nonlinear ordinary differential equation was carried out, and subsequently, the resulting equations were solved using the Jacobi elliptic function method. The construction of exact solutions, including solitary wave, periodic, and shock wave solutions, for the fractional order of the Gardner equation was performed. A comparison between the exact solutions and the fractional solutions was presented. This work is important because the suggested technique offers a simple and efficient way to examine a wide range of nonlinear fractional differential equations. By employing this approach, it becomes possible to solve several nonlinear time-fractional differential equations that involve conformable derivatives. The graphical representation of the resulting data simplifies the process of determining the physical significance of the equation. [ABSTRACT FROM AUTHOR]

In this study, we solve by involving conformable fractional derivative on time-fractional perturbed nonlinear Schrödinger model which describe the behaviour of soliton transmission on fibers optical system in physics specially in transmission of data over long distances with large bandwidth. There are two modern techniques are utilized for solving the suggested model namely the Extended hyperbolic function technique and the polynomial expansion technique. These techniques give unique, robust, and powerful solutions that are useful in many research areas. We attain different types of solutions that give unique behaviour of V shaped, singular solution, and periodic soliton solutions. These solutions are play a vital role in the soliton theory along with data transmission in optical fibers. Further we also discuss the stability of obtain solutions as well as the modulation instability of the governing NLSE. To grasp and better understanding of the solution behavior we include the 3D, contour and 2D graphics. [ABSTRACT FROM AUTHOR]

In optical fibers, the generalized nonlinear Schrödinger equations with self-steepening (SS), self-frequency shift (SFS), intermodal dispersion (IMD), and third-order dispersion (TOD) play an important role. Our investigation covers a variety of physical parameters based on how optical solitons change their structure as they move through an optical medium. Our study shows that modifying the coefficients for SS, SFS, IMD, and TOD can affect optical solitons' profiles either by altering their nature or without doing so. We used the extended rational sinh–cosh method, which works with various types of soliton profiles. These profiles include dark, kink-dark, kink, and anti-kink solitons. By selecting appropriate physical parameter values, the behavior of various optical solitons is graphically depicted. As a result, we utilize the eigenvalue spectrum to investigate linear stability analysis. [ABSTRACT FROM AUTHOR]

The space–time fractional Biswas–Arshed and Schrödinger Kerr law equations featuring beta derivative hold substantial application in nonlinear optics, optical solitons, ultrafast optical signal, nonlinear photonics, quantum optics, biophotonics, photonic crystals photonics, etc. In this study, a wide variety of geometric shape solitons have been established that include hyperbolic, exponential, trigonometric, and rational functions, as well as their assimilation to the considered equations, through the two-variable ( R ′ / R , 1 / R )-expansion approach. The implication of the fractional parameter μ on the wave shape has also been examined by depicting two-dimensional and three-dimensional plots for particular parameter values. The solitons include irregular periodic, pulse like, V-shaped, bell-shaped, positive periodic, asymptotic, general solitons, and some others. It is significant to note that the changes in the wave pattern result from the adjustments to substantive and auxiliary parameters. The outcomes demonstrate the efficiency, acceptability, and dependability of the ( R ′ / R , 1 / R )-expansion approach for obtaining solutions to the fractional-order evolution equations in the domains of engineering, technology, and sciences. It is evident from the graph that changing the value of μ results in a change in the shape of the soliton. The study explores how these equations change as fractional-order derivatives vary. Soliton solutions, which are stable, localized waveforms, are crucial in optical communication systems. Understanding their behavior under changing fractional-order derivatives is essential for advancing optical signal processing and communication technologies. [ABSTRACT FROM AUTHOR]

In this study, the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation (KP-BBME) is examined. KP-BBM is used as a water wave model to mimic the wave propagation for fluid flows and to describe bidirectional propagating water wave surface. Studying is conducted by applying the modified extended mapping method to construct various and novel solutions for the proposed model. These solutions including {dark, bright, and singular} solitons, Weierstrass elliptic, exponential and singular periodic solutions. The extracted solutions confirmed the efficacy and strength of the current technique. To illustrate the physical characteristics of the established solutions, 3D, 2D and contour plots are depicted for many selected solutions. [ABSTRACT FROM AUTHOR]

This article reveals the different types of optical solitons of non-linear coupled Riemann wave equation and Wazwaz Kaur Boussinesq equation. We adopted a direct integration technique namely, modified G ′ G 2 -expansion. Different sorts of soliton's existence criteria are also presented here. The proposed technique provides the new travelling wave solutions with the aid of different types of derivatives such as beta derivative, M-Truncated derivative and Conformable derivative and also offers special kinds of solutions including rational, trigonometric and hyperbolic solutions. In this work, we compared and analysed solitary wave solutions obtained by using different types of fractional derivatives. The outcomes of the study are highly significant for modern communication network technology, optical fiber, ion-acoustic, magneto-sound waves in plasma, and stationary media, particularly in the propagation of tidal and tsunami waves. [ABSTRACT FROM AUTHOR]

In this article, we apply the extended modified auxiliary equation mapping method to study the generalized Nizhnik–Novikov–Veselov equation,the Radhakrishnan–Kundu–Lakshmanan equation, and a cold bosonic atom in a zig–zag optical lattice. By employing this method, we obtain soliton solutions of different types and utilize computer assistance to visualize these solutions. [ABSTRACT FROM AUTHOR]

The major goal of the current research is to investigate the effects of fractional parameters on the dynamic response of soliton waves of fractional non-linear density-dependent reaction diffusion equation. Two well-known integration methodologies: the advanced exp (- Θ (ξ)) -expansion method and the modified auxiliary equation method in the sense of beta derivative and M-fractional derivative have been implemented to achieve explicit solitonic solutions of the fractional non-linear density-dependent reaction diffusion equation that emerged in mathematical biology. The spatial dynamics of populations, chemical concentrations, or other quantities are commonly studied using this equation type in biology, ecology, and chemistry. Solitary wave solutions of the governing equation, representing the dynamics of waves, plays a vital rule in many branches of biology, ecology, and chemistry. The obtained solutions has been studied in the form of singular kink-type solitary wave and kink-wave solutions. The behavior of soliton wave solutions is also demonstrated via 2D and 3D graphs. As a result of the fractional effects, physical changes are observed. The acquired results manifest that the proposed methods are more convenient, adequate, powerful and efficacious than other direct analytical methods. The attained results might improve our understanding of how waves propagate and could benefit the fields of medicine and allied sciences. [ABSTRACT FROM AUTHOR]

This study focuses on investigating analytical soliton solutions within the context of the time-fractional cubic-quintic nonlinear non-paraxial pulse propagation model, an adaptive model with extensive uses in various intricate real phenomena, such as nonlinear optics, fiber optics communication, optical signal processing, laser-tissue interaction in biomedical science, among others. Depending on the strength of the cubic and quintic nonlinear terms, various nonlinear effects, including self-focusing, self-phase modulation, and wave mixing, can be observed. This model is investigated using a powerful analytical technique, the (G ′ / G , 1 / G) -expansion approach, which develops several potential solitons that allow for insight into the laser pulse interactions. This study yields diverse illustrative soliton solutions, including periodic, bell-shaped, kink, singular solitons, etc. some of which have been documented in former literature. Furthermore, we conduct an extensive analysis of these solitons, considering both anomalous and normal group velocity dispersion, and effectively visualize the results through two- and three-dimensional graphs, along with contour plots. The findings in this article could hold significance for researchers engaged in the advancement of optical equipment, biomedical laser devices, mode-locked lasers, and similar technologies. [ABSTRACT FROM AUTHOR]

The exact solutions of the (2 + 1) dimensional Broer–Kaup–Kupershmidt (BKK) system which has been recommended to model the nonlinear and dispersive long gravity waves traveling along with the two horizontal directions in the shallow water of uniform depth were obtained. Firstly, the given system was reduced to an ordinary differential equation (ODE) with the help of the wave transformations. Then, the reduced ODE was solved with the help of two methods which are called the modified (G ′ / G) -expansion method and new extended generalized Kudryashov method. We checked the results with the Maple software and plotted 3D, contour and 2D plots of some obtained solutions. As a result, we obtained exact solutions that are different from each other and have not been obtained before. Results can enhance the nonlinear dynamical behavior of a given system and demonstrate the effectiveness of the employed methodology. Results will be beneficial to a large number of engineering model specialists and useful for understanding the wave motions. [ABSTRACT FROM AUTHOR]

In this work, the fractional coupled nonlinear Schrödinger model (FCNLSM) in optical fibre can exhibit chaotic dynamics with potential applications in cryptography and applied mathematics. This study investigates the dynamical properties of FCNLSM in optical fibre. The study depends on four steps. (1): The governing model is converted into an ordinary differential equation by employing a wave transformation. The soliton solutions not discussed in the literature are obtained by employing the new direct algebraic method. (2): The governing model is simplified into a dynamical system (DS). Through phase plane analysis, both perturbed and unperturbed behaviour of FCNLSM is observed. (3): The chaotic behaviour in the perturbed DS is characteristic and is harnessed for the design of an efficient encryption algorithm. The proposed encryption technique involves computing the hash digest of a plain image, which is then used to update the initial seed of the chaotic FCNLSM. The one-way function is employed to ensure the security of the encryption method. (4): The encryption and decryption coding is applied to confuse and diffuse the plain image, followed by decoding to generate the cypher image. The security performance of the proposed cryptosystem is evaluated using established metrics, demonstrating its resilience against various cryptanalysis techniques. Moreover, complexity analysis indicates the feasibility of practical implementation for the proposed algorithm. [ABSTRACT FROM AUTHOR]

The Kundu-Mukherjee-Naskar equation holds significant relevance as a nonlinear model for investigating intricate wave phenomena in fluid and optical systems. This study uncovers new optical soliton solutions for the KMN equation by employing analytical techniques that utilize combined elliptic Jacobian functions. The solutions exhibit mixtures of distinct Jacobian elliptic functions, offering novel insights not explored in prior KMN equation research. Visual representations in the form of 2D ContourPlots elucidate the physical behaviors and properties of these newly discovered solution forms. The utilization of symbolic computations facilitated the analytical derivation of these solutions, offering a deeper understanding of the nonlinear wave dynamics governed by the KMN equation. These employed techniques showcase the potential for future analytical advancements in unraveling the complex soliton landscape of the multifaceted KMN model. The findings provide valuable insights into the intricacies of soliton behavior within this nonlinear system, offering new perspectives for analysis and exploration in areas such as fiber optic communications, ocean waves, and fluid mechanics. Maple symbolic packages have enabled us to derive analytical results. [ABSTRACT FROM AUTHOR]

In order to obtain the local structural solutions of nonlinear integrable systems, a (3 + 1)-dimensional nonlinear integrable equation is studied by using the multi-linear variable separation method, and the soliton, dromion, breather and instanton solutions containing arbitrary functions are obtained. Then, the abundant local excitations for the proposed equations are constructed by appropriately setting arbitrary function forms, and the evolution characteristics of system's dromion solutions with time are investigated. In addition, the fractal structure of the separable solution of the system was described. The results show that the proposed method can obtain some special solutions and this method has been extended in different ways so as to enroll more low-dimensional functions in the solution. [ABSTRACT FROM AUTHOR]

This article pays attention to secure the exact wave structures to the system of ion sound under influence of ponderomotive force. It is caused by non-linear force and is experienced by a charged particle in an oscillating electromagnetic field of inhomogeneity. The studied equation is analyzed by the assistance of conformable time-fractional and composed of normalized density perturbation and normalized electric field of the Langmuir oscillation. The solutions are extracted with the assistance of new extended direct algebraic method, a relatively new integration tool. We extract various wave structure in solitons in different forms like, bright, dark, combo, and singular soliton solutions. In addition to being helpful for elucidating FNLPDEs, the technique both returns solutions that have already been retrieved and generates new exact solutions. Assuming appropriate parameter values, a variety of graph shapes are sketched to describe the graphical presentation of the calculated outcomes. We anticipate that many engineers who use engineering models will find this study to be of interest. The results demonstrate the viability, ease of use, and scalability of the selected computational approach, even when applied to complex systems. [ABSTRACT FROM AUTHOR]

The fractional coupled nonlinear Schrödinger model (FCNLSM) is widely utilized in various fields such as nonlinear optics, optical communication systems, plasmas and mathematical physics. In this study, we aim to achieve three primary objectives. Firstly, we seek to obtain novel soliton solutions for the FCNLSM, which have not been previously reported in the literature. Secondly, we employ the Sardar sub-equation method and the improved generalized tanh-function method to effectively analyze the dynamics of solutions and solve the studied model. These methods provide valuable insights into the behavior of the system. Lastly, we conduct bifurcation and stability analyses to explore the dynamical properties of the model. To ensure the physical validity of our findings, we present 2-dimensional, 3-dimensional and contour plots using carefully selected parameter values. The obtained results demonstrate the feasibility, efficiency and computational speed of the employed techniques in obtaining comprehensive and reliable solutions. The study represents a novel and significant contribution to the field by expanding the understanding of soliton solutions in the FCNLSM, introducing new techniques for their investigation and conducting a comprehensive analysis of bifurcation and stability properties. The findings of this research open new avenues for exploration and application in the areas of nonlinear optics, optical communication systems and other related fields. [ABSTRACT FROM AUTHOR]

This study investigates the fractionally coupled nonlinear Schrödinger model (FCNLSM), which has numerous applications in different fields of physics, such as optics, condensed matter physics and plasma physics. The study employs two versatile techniques, the unified technique and the modified F -expansion technique, to explore various solutions. By applying these techniques, we obtain novel soliton solutions, which are expressed in terms of rational, hyperbolic and trigonometric solutions, along with kink, periodic and singular soliton solutions. Additionally, multi-wave U-shaped solitary wave solutions are assessed. The sensitivity analysis of the model is investigated and distinctive 2-dimensional, 3-dimensional and density graphs are used to illustrate the behavioral characteristics of the retrieved solutions. As far as we know, this manner of investigation has never been explored before. The results demonstrate the reliability, consistency and effectiveness in finding precise solutions to the various difficult nonlinear issues that arise in engineering, applied sciences and nonlinear optics. [ABSTRACT FROM AUTHOR]

In this work, we proposed a new modified generalized Riccati equation mapping approach to successfully extract several analytical soliton solutions for the (2+1)-dimensional nonlinear Schrödinger (NLS) equation with the help of symbolic computation works in Mathematica.The (2+1)-dimensional NLS equation is used in many fields, including plasma physics, nonlinear optics, and quantum electrodynamics. The main objective of the present work is to develop an effective methodology for solving highly nonlinear evolution equations that are influenced by the enhancement of a previously known method. The approach under consideration is a newly improved version of the classic generalized Riccati equation mapping. By taking advantage of this newly proposed method, we produced a wide range of closed-form solutions, including new optical solitons, traveling waves, and soliton-like solutions, all of which are crucial for nonlinear optics, optical fibers, and the physical propagation of electromagnetic waves. One may clearly argue that the novel method is highly effective and successful in finding exact solutions to nonlinear evolution equations. Moreover, we obtained a variety of new families of soliton-like wave solutions. By using the mathematical software Mathematica, we also created 2D, 3D, and contour graphics for some of the reported solutions by choosing suitable parameter values. [ABSTRACT FROM AUTHOR]

In this article, a seventh-order variable-coefficient nonlinear Schrödinger equation is investigated in an optical fiber. By means of the Darboux transformation, soliton and breather solutions are derived and the following results are attained: (i) The one soliton and interactions of two solitons are presented, whose basic structures are parabolic-like, cubic and periodical-oscillating solitons; (ii) The first-order breather and interactions between the two breathers are studied. Several interesting nonlinear wave patterns such as cow-shaped breathers and breathers with periodic properties are displayed; (iii) The dynamic behaviors of solitons and breather waves are affected by the variable coefficients. [ABSTRACT FROM AUTHOR]

The paper is intended to investigate of the extended time fractional (2+1) dimensional Zakharov–Kuznetsov (tf-ZK) equation with the sense of Riemann–Liouville (R–L) fractional derivative operator. Utilizing the planar dynamical system theory, determined the equilibrium points for different cases and properties revealed. We also show phase portraits of propagating wave solutions for taking the some parameters with special values. At the same time, we employ the fractional complex transformation for the generalized exponential rational function method (GERFM) and sub equation method, we construct abundant exact solitary wave, kink wave, refracted wave and periodic wave solutions. Taking the special values for some parameters interesting figures of traveling wave solutions were depicted. [ABSTRACT FROM AUTHOR]

To study the motion of microscopic particles, the Fokas–Lenells model is a completely integrable equation that describes the profile of light pulses propagating through nonlinear mono-mode optical fibers. Moreover, the suggested equation has wide applications in geophysics, sound wave propagation in nonhomogeneous media, fluid mechanics, and mathematical modeling of various phenomena, such as traffic flow, financial market, etc. This study computes several inclusive and standard optical soliton solutions to the space–time fractional Fokas–Lenells equation in the beta derivative sense using the reputed (G ′ / G , 1 / G) -expansion approach. Furthermore, for different values of fractional orders, we analyze the impact of fractional derivative through two- and three-dimensional graphical representations of the solitons. The stated approach determines the analytical soliton solutions with geometric shapes, like kink, periodic soliton, bell-shaped, kink-periodic, V-shaped, and some other solitons to the Fokas–Lenells equation. The obtained solutions are compared with former solutions, which demonstrate the competence of the chosen approach in providing apposite solutions. [ABSTRACT FROM AUTHOR]

This study employs the generalized exponential rational function method and ( G ′ G 2 )-expansion function method. Computer algebra is used to study the multiple wave solutions to the (2+1)-dimensional nonlinear Zakharov–Kuznetsov modified equal-width problem. In the study of plasma physics, this model is utilized to represent the effects of a magnetic field on a weak ion-acoustic waves. Bright, dark, singular, and their combo forms solutions are extracted. Besides, the different kinds of the hyperbolic, trigonometric, rational, exponential function and singular periodic wave structures are also obtained. The parameter constraints for the existence of such solitons are also enumerated. With suitable parameter values, the results are shown and supported theoretically by visualizing 3D surface plots, 2D line plots, and corresponding contour graphs. The results of this research show that the methods used to improve the system's nonlinear dynamical behavior are effective. Intricate phenomena can now be analyzed with computational tools that are not only effective but also easy to use and compatible with one another. We believe that many engineering model professionals will find this work useful as well. Mathematica has been used to confirm the accuracy of these solutions for all retrieved results. [ABSTRACT FROM AUTHOR]

In this research, we concentrated on the dispersive long wave system in two horizontal directions for dispersive nonlinear waves on the shallow water of an open sea or a wide channel of finite depth. We investigated this governing system by using two different methodologies, namely the generalized exponential rational function (GERF) method, and the new modified generalized exponential rational function (MGERF) method. The GERF method was first introduced by Ghanbari and Inc (Eur. Phys. J. Plus 133:-142, 2018) for finding the soliton solutions for highly nonlinear partial differential equations (NLPDEs). This technique is very reliable and straightforward and reduces the NLPDEs into ordinary differential equations (ODEs) under the wave transformation. Being motivated by the GERF technique, we proposed a newly modified generalized exponential rational function (MGERF) method under wave transformation. We obtained a diverse set of solutions involving trigonometric forms, hyperbolic forms, rational forms, and so on, which have a broad application spectrum in fields such as plasma physics, nonlinear optics, optical fibers, and nonlinear sciences by utilizing these methods. Due to the presence of various arbitrarily chosen constants, these solutions exhibit extensive and rich dynamical behavior. Based on the dynamical behaviors, we discovered that the soliton solutions were collisions of solitons, breather-like solitons, line-form solitons, multi-solitons, solitary waves, lump-form solitons, and other forms. Consider a nonlinear system with dispersive and dispersion terms, a nonlinear Schrödinger equation, and fractional nonlinear evolution equations, which will yield additional interesting and more achievable results. Finding solutions to these equations in solitary wave solution forms will be a difficult task. [ABSTRACT FROM AUTHOR]

We have extracted some soliton solutions of the fractional longitudinal wave equation with the M-truncated derivative (M-LWE), which emerges in a magneto electro-elastic circular rod. To obtain new results of this model, the unified Riccati equation expansion and new Kudryashov methods have been utilized for the first time. The presented methods have been productively implemented to the considered model. With the help of these two methods, new soliton waves of the M-LWE have been obtained successfully. For a better understanding of the subject and analysis of the results, 3D, contour, and 2D graphs of some soliton solutions have been presented. The interesting part of our work is that both methods, named unified Riccati equation expansion and the new Kudryashov methods have successfully been applied for the first time to get new soliton solutions of M-LWE. [ABSTRACT FROM AUTHOR]

The present research focuses on fractional nonlinear evolution equations and their optical soliton solutions, which have become the inquisitive context to study their significant attributes in understanding natural kernels ascending in the field of science and technology. This article has been dedicated to searching out the analytical soliton solution of an important fractional nonlinear evolution equation, named the time-fractional Kundu–Eckhaus equation in the sense of beta fractional derivative through the ( G ′ / G , 1 / G )-expansion approach. This equation was originated to search out the transmission of data through the optical fiber. By exerting the stated method, abundant novel soliton solutions, like kink soliton, compacton, periodic soliton, singular periodic, singular bell-shaped soliton, and others have been established. In accordance with the trail solutions generated in this method, the solutions contain arbitrary parameters and hyperbolic, rational, and trigonometric functions. Soliton solutions are extracted from analytical solutions for apposite values of the parameters. Contour, three- and two-dimensional graphs are plotted to demonstrate the physical structure and characteristics of the attained solitons. The obtained results imply that the concerned method can be used to attain diverse, improved, useful, and compatible solutions for other significant fractional nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This analysis utilizes the generalized Riccati simple equation method to construct nematicons in liquid crystals from its governing system. A new type of nonlinearity is studied for the first time in the context of liquid crystals. It is the nonlinear quadruple power law. The fractional version of the governed model, with conformable sense, is considered. With the aid of Mathematica, Bright, dark, and singular types of solutions are derived with the constraints guaranteeing their existence. Moreover, some obtained results are depicted to show the real physical characteristics of nematicons. The used method provides a powerful tool for extracting exact solitary wave solutions. By simple calculations, we show that the generalized Riccati simple equation method can be treated as a general case of the well-known simple equation, exp - φ (η) -expansion, and G ′ / G -expansion methods. [ABSTRACT FROM AUTHOR]

This study presents new soliton solutions for the nonlinear fractional twin-core couplers with Kerr law nonlinearity by employing the modified extended tanh method with Riccati equation. The solutions are expressed in terms of some elementary functions including rational, trigonometric and hyperbolic types. Graphical demonstrations of the simulations are provided. [ABSTRACT FROM AUTHOR]

The ( G ′ G) -expansion and exponential rational function methods (ERFM) are proposed for generating the precise solutions of a generalized coupled integrable dispersionless system that occurs in the study of a variety of problems in applied mathematics and physics. The current study validates the key characteristics of the techniques used, and precise kink solutions are obtained using the established procedures. [ABSTRACT FROM AUTHOR]

In this article, the new exact solitary wave solutions for the generalized nonlinear Schrödinger equation with parabolic nonlinear (NL) law employing the improved tanh (Γ (ϖ)) - coth (Γ (ϖ)) function technique and the combined tan (Γ (ϖ)) - cot (Γ (ϖ)) function technique are obtained. The offered techniques are novel and also for the first time in this study are used. Different collections of hyperbolic and trigonometric function solutions acquired rely on a map between the considered equation and an auxiliary ODE. The several hyperbolic and trigonometric forms of solutions based on diverse restrictions between parameters involved in equations and integration constants that appear in the solution are obtained. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model utilizing mathematical software. The main subject of this work is that one can visualize and update the knowledge to overcome the most common techniques and defeat to solve the ODEs and PDEs. The concluded solutions are demonstrated where are valid by using Maple software and also found those are correct. The proposed methodology for solving the metamaterilas model are designed where is effectual, unpretentious, expedient, and manageable. Finally, the existence of the obtained solutions for some conditions is also analyzed. [ABSTRACT FROM AUTHOR]