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We study the asymptotic stability of peaked solitons under H1 × H1-perturbations of the two-component Novikov equation involving interaction between two components. This system, as a two-component generalization of the Novikov equation, is a completely integrable system which has Lax pair and bi-Hamiltonian structure. Interestingly, it admits the two-component peaked solitons with different phases, which are the weak solutions in the sense of distribution and lie in the energy space H1 × H1. It is shown that the peakons are asymptotically stable in the energy space H1 × H1 with non-negative momentum density by establishing a rigidity theorem for H1 × H1-almost localized solutions. Our proof generalizes the arguments for studying the Camassa-Holm and Novikov equations. There are three new ingredients in our proof. One is a new characteristic describing interaction of the two-components; the second is new additional conserved densities for establishing the main inequalities; while the third one is a new Lyapunov functional used to overcome the difficulty caused by the loss of momentum. [ABSTRACT FROM AUTHOR]

In this work, we take into account the (2+1)-Davey Stewartson equation (DSE) and the (2+1)-complex coupled Maccari system (CCMS) and their analytical solutions. Besides, we tackle the role of the problem parameters on the soliton behavior produced by the presented DSE. Exact traveling wave solutions are highly useful in numerical and analytical theories for such equations. While numerical methods are widely used, improving analytical approaches for obtaining analytical solutions is necessary for a deeper understanding of dynamics. This study marks a significant milestone by implementing the efficient analytical approach, enhanced modified extended tanh expansion method, for the first time to the (2+1)-DSE and (2+1)-CCMS equations, thereby making a notable contribution to the existing literature. We have shown that features of the soliton solutions can represent the spread of propagation on the wavefronts and show a reasonable dependency on parameter values. Some of the solutions discovered in three- and two-dimensional arrangements can also be described in graphic representations of their behavior. With the help of the graphical depictions, bright, singular, and periodic singular soliton characters for the (2 + 1) -DSE and singular, dark, bright, and periodic singular soliton characters for the (2 + 1) -CCMS are acquired. The results show that the utilized analytical technique is easily applicable, efficient, reliable, robust, and categorical when it comes to finding analytical solutions for different nonlinear models. Moreover, the problem parameters and the coupling coefficients have significant influences on the behavior of the solitons of the DSE, and this examination is studied for the first time in this article. [ABSTRACT FROM AUTHOR]

In an optical fiber, we report on our study of pulse compression using an analytical method. A significant role for the eighth order nonlinear Schrödinger (NLS) equation may be seen in the study of ultra-short pulses, in particular extremely nonlinear optical phenomena. As a result of our study, which plays a significant role in reviving potent mathematical procedures, such as the extended rational sinh–cosh method for obtaining the dark soliton solution and the cosine method for solving the NLS equation to attain M-shaped, W-shaped soliton structure, dark and bright solitonic structure. Solitons are shown to have a strictly chirp-free structure, which facilitates effective compression. According to the results, wavevector may effectively regulate the shape and propagation behavior of soliton. [ABSTRACT FROM AUTHOR]

In this work, we implement the bifurcation theory for the planar dynamical system and the complete discrimination system method of polynomials to the integrable Kuralay equations. Solitons generated by these equations are found to be relevant in diverse fields such as ferromagnetic materials, nonlinear optics, and optical fibers. Through specific wave transformations, it undergoes conversion into an ordinary differential equation (ODE), which is subsequently transformed into a planar dynamical system associated with a one-dimensional Hamiltonian function. As per the qualitative theory of the planar dynamical system, phase portraits of the Hamiltonian system are plotted and used to construct some new traveling wave solutions. Numerical examination reveals diverse nonlinear structures in the analytical solutions, encompassing solitary waves, kink waves, and periodic wave profiles. Additionally, the integrable Kuralay equation employs the complete discrimination system method of polynomial for the first time, yielding solutions expressed in trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. Visual representations of the derived solutions include 3-D, 2-D, and contour plots. The reliability and effectiveness are affirmed through the numerical graphs of the solutions. Furthermore, we conducted a numerical investigation into the chaotic and quasiperiodic behavior of the perturbed system by introducing a specific periodic force into the primary system. [ABSTRACT FROM AUTHOR]

In this paper, the (2 + 1)-dimensional variable coefficients equation which describes the thermophoric wave motion of wrinkles in graphene sheets (2D-vGS) is studied, where it has many applications in 2D optics, nanophotonic, and nanoelectronics. A direct simplified Hirota's bilinear method is generalized to find the bilinear form of the 2D-vGS equation. Accordingly, one, two, and three soliton wave solutions indicate that our studied equation is fully integrable and has n-soliton solutions. Moreover, we have focused on the study of two and three solitons interactions, this leads to the identification of two distinct solution types, the Y-shape soliton and fork- shape soliton, which can be clearly distinguished from the 3D plots and density plots. These solutions are characterized by a rich spectrum of collision dynamics and encompassing phenomena such as fusion and fission. The nonlinear properties of the two and three soliton solutions could be useful for farther applications in 2D optics like metamaterials with exotic optical properties and ultra-compact and efficient photonic devices. [ABSTRACT FROM AUTHOR]

This research investigates the utilization of a modified version of the Sardar sub-equation method to discover novel exact solutions for the generalized Pochammer Chree equation. The equation itself represents the propagation of longitudinal deformation waves in an elastic rod. By employing this modified method, we aim to identify previously unknown solutions for the equation under consideration, which can contribute to a deeper understanding of the behavior of deformation waves in elastic rods. The solutions obtained are represented by hyperbolic, trigonometric, exponential functions, dark, dark-bright, periodic, singular, and bright solutions. By selecting suitable values for the physical parameters, the dynamic behaviors of these solutions can be demonstrated. This allows for a comprehensive understanding of how the solutions evolve and behave over time. The effectiveness of these methods in capturing the dynamics of the solutions contributes to our understanding of complex physical phenomena. The study's findings show how effective the selected approaches are in explaining nonlinear dynamic processes. The findings reveal that the chosen techniques are not only effective but also easily implementable, making them applicable to nonlinear model across various fields, particularly in studying the propagation of longitudinal deformation waves in an elastic rod. Furthermore, the results demonstrate that the given model possesses solutions with potentially diverse structures. [ABSTRACT FROM AUTHOR]

This research paper's primary goal is to find fresh approaches to the Hirota–Maccari system. This system explains the dynamical features of the femto-second soliton pulse in single-mode fibers. The bright soliton, dark soliton, dark-bright soliton, dark singular, bright singular, periodic soliton, and singular solutions are developed utilizing the modified auxiliary equation technique. To make the physical significance of each unique solution clearer, it is mapped in both 2D and 3D. The primary Hirota–Maccari system is being verified by all new solutions, and the constraint condition is also provided. The obtained optical solitons may be important for the analysis of nonlinear processes in optic fiber communication and signal processing. [ABSTRACT FROM AUTHOR]

This current research explores a new generalized q-deformed sinh -Gordon equation. The governing equation is frequently used to model plasma-based solutions, complex optical science, photonic transmission systems, ultrashort burst lasers to identify objects and other theoretical sciences. The work consists of two goals. Firstly, analytically using the modified extended tanh function approach. Precise results for the derived equation are obtained that could be used to simulate physical systems with broken symmetries and to consider events involving amplification or dissipation. Secondly, with the help of a numerical approach investigate the error function between the q-demormed terms. Several figures have been included to illustrate the different optical solitons propagation patterns. [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]

Nonlinear dynamical problems, characterized by unpredictable and chaotic changes among variables over time, pose unique challenges in understanding. This paper explores the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation-a fundamental equation in applied magnetism and nanophysics. The study focuses on dynamic behaviors, specifically examining bifurcations and equilibrium points leading to chaotic phenomena by introducing an external term to the system. Employing chaos theory, we showcase the chaotic tendencies of the perturbed dynamical system. Additionally, a sensitivity analysis using the Runge-Kutta method reveals the solution's stability under slight variations in initial conditions. Innovatively, the paper utilizes the planar dynamical system technique to construct various solitons within the governing model. This research provides novel insights into the behavior of the (2+1)-dimensional cmKdV equation and its applications in applied magnetism and nanophysics. [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 deals with a new variant of the Biswas–Milovic equation, referred to as the perturbed Biswas–Milovic equation with parabolic-law nonlinearity in spatio-temporal dispersion. To our best knowledge, the considered equation which models the pulse propagation in optical fiber is studied for the first time, and the abundant optical solitons are successfully obtained utilizing the auxiliary equation method. Utilizing a wave transformation technique on the considered Biswas–Milovic equation, and by distinguishing its real and imaginary components, we have been able to restructure the considered equation into a set of nonlinear ordinary differential equations. The solutions for these ordinary differential equations, suggested by the auxiliary equation method, include certain undetermined parameters. These solutions are then incorporated into the nonlinear ordinary differential equation, leading to the formation of an algebraic equation system by collecting like terms of the unknown function and setting their coefficients to zero. The undetermined parameters, and consequently the solutions to the Biswas–Milovic equation, are derived by resolving this system. 3D, 2D, and contour graphs of the solution functions are plotted and interpreted to understand the physical behavior of the model. Furthermore, we also investigate the impact of the parameters such as the spatio-temporal dispersion and the parabolic nonlinearity on the behavior of the soliton. The new model and findings may contribute to the understanding and characterization of the nonlinear behavior of pulse propagation in optical fibers, which is crucial for the development of optical communication systems. [ABSTRACT FROM AUTHOR]

The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is investigated in this study. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is an extension of Calogero–Bogoyavlenskii–Schiff equation that describes the movement of Riemann waves along y-axis while long waves moves along the x-axis. The dynamics of Riemann waves is one of the most significant applications including tsunami in rivers, internal waves in oceans and magento-sound waves in plasmas. Finding new precise solutions with the assistance of a relatively new extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique is the primary objective of this effort. The suggested techniques are important tools in the fields of mathematical physics. Successful extraction of hyperbolic, rational, and trigonometric function solutions are achieved by using the proposed analytical methods. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is studied for the first time using extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique in this work and novel solutions are observed. 3D plots, contour plots and 2D plots are used to depict the dynamics of the extracted 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]

The central objective of this study is to explore the dynamic response of fractional-soliton solutions within a nonlinear T z i t z e ´ i c a –Dodd–Bullough (TDB) equation enhancing the long-distance optical communication, developing advanced materials with unique electromagnetic properties, and contributing to a deeper understanding of complex phenomena. This fractional version integrates fractional derivatives to facilitate the modeling of anomalous diffusion and various other non-local phenomena. We approach the governing model using the extended direct algebraic method, leading to the derivation of fractional-soliton solutions. These solutions are not only exhibited but also have their physical implications elucidated, with two fractional derivative definitions serving as the interpretive tools: the β -derivative and a novel local derivative. The aforementioned integration approach enables the derivation of numerous modern optical soliton solutions, encompassing dark, semi-bright, as well as solutions involving trigonometric, mixed hyperbolic, rational functions, and dark singular solitons. This method effectively highlights the fractional impact of the derived physical phenomena on the fTBD equation. Additionally, the fractional dynamical system undergoes a thorough sensitivity analysis, with the results being graphically represented. To facilitate this, the model undergoes transformation into a planar dynamical system via the Galilean transformation, allowing for an evaluation of the sensitivity performance. [ABSTRACT FROM AUTHOR]

A method is presented for the construction of W shaped-W shaped solitons and W-shaped soliton combined with other types of solitons to coupled nonlinear partial differential equations (NLPDEs). This method is applied to investigate the propagation of new progressive solitons in magneto-optical waveguides that carry the quadratic-cubic nonlinearity described by a coupled system of nonlinear Schr o ¨ dinger equations. These waveguides have a great advantage since they control solitons' clutter effects and thus assure their smooth propagation across intercontinental distances. Many approaches to investigating the exact solitons and other solutions to nonlinear models are often very powerful and very efficient to deal with single NLPDEs, however they do not always have the same success when applied to coupled NLPDEs. This difficulty is due to the existence of interaction terms in the coupled equations. For this reason, it is not always an easy task to derive exact solutions to these coupled equations. To overcome this difficulty, several authors have attempted to solve coupled equations by making the assumption that the solution in one line is proportional to the solution in another line, leading to an excessive imposition of constraints. This leads to the reduction of coupled equations to a single one. By doing so, they denature the physical phenomena described by the original coupled NLPDEs. We propose here a method that gives a better perspective in handling the analytical investigation of coupled NLPDEs. The novelty lies on the fact that it not only allows the propagation of different types of solutions in two lines to be conducted in just one move, but it also retrieves the propagation of same types of solutions in the two lines. For this latter case, the solutions come naturally with less constraints.The criteria for these solutions to exist are also exhibited in the form of parameter constraints. [ABSTRACT FROM AUTHOR]

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

The thermophoretic motion (TM) system with a variable heat transmission factor, based on the Korteweg-de Vries (KdV) equation, is used to model soliton-like thermophoresis of creases in graphene sheets. Painlevé test is employed to discover that the equation is Painlevé integrable. Then an auto-Bäcklund transformation using the truncated Painlevé expansion is obtained. Concerning the additional variables, the auto-Bäcklund transformations convert the nonlinear model to a set of linear partial differential equations. Finally, various explicit precise solutions based on the acquired auto-Bäcklund transformations are investigated and the researched solutions are illustrated in 3D, 2D and contour plots. Furthermore, the Cole-Hopf transformation is used in conjunction with Hirota's bilinear technique to get multiple regular and singular kink solutions. [ABSTRACT FROM AUTHOR]

In this study, the beta time fractional (2 + 1) dimensional Chaffee–Infante equation used to describe the behavior of gas diffusion in a homogeneous medium is discussed. Generalized Kudryashov and modified Kudryashov procedures were used to discovered solitons of the equation. These methods can be easily applied and offer different solutions checked to other methods in the literature. At the same time, these two methods use symbolic calculations to better understand various nonlinear wave models and offer a powerful and effective mathematical approach. The solutions created in this article are different from those in the literature and will guide those working in the field of physics and engineering to better understand this model. Figures of the results were made values different from each other. The stability of the equations in applications has been demonstrated by testing the stability feature on some solutions obtained using the features of the Hamilton system. This work demonstrates the power and effectiveness of the methods discussed in applying many different forms of fractional-order nonlinear equations. The results obtained in this paper are original to our research and have the potential to be helpful in the fields of mathematical engineering and physics. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble. [ABSTRACT FROM AUTHOR]

This work is on the retrieval of quiescent optical solitons for the complex Ginzburg–Landau equation that is with nonlinear chromatic dispersion and generalized structure of quadratic–cubic form of self–phase modulation. The Lie symmetry is applied to make this retrieval possible. The model is studied with linear temporal evolutions as well as generalized temporal evolution. [ABSTRACT FROM AUTHOR]

The (3 + 1)‐dimensional Vakhnenko–Parkes mathematical model has a wide range of applications in science and engineering. In this paper, the model studied is investigated and analyzed by using two effective schemes such as sine‐Gordon expansion method and its newly developed version, rational SGEM. Moreover, many novel properties of model studied are extracted in detail. Furthermore, their wave distributions properties and graphs are also plotted under the strain conditions. Interactions of the gravitational potential and high‐frequency wave perturbation properties are also reported in a detailed manner. [ABSTRACT FROM AUTHOR]

This paper initiates an exploration into the exact solutions of the variable coefficient Date-Jimbo-Kashiwara-Miwa equation, first utilizing the Painlevé analysis method to discuss the integrability of the equation. Subsequently, By employing the Hirota bilinear method, N-soliton solutions for the equation are constructed. The application of the Long wave limit method to these N-soliton solutions yields rational and semirational solutions. Various types of localized waves, encompassing solitons, lumps, breather waves, and others, emerge through the careful selection of specific parameters. By analyzing the image of the solutions, the evolution process and its dynamical behavior are studied. [ABSTRACT FROM AUTHOR]

The main aim of this study is to conduct an in-depth exploration of a recently introduced extended variant of the Kadomtsev–Petviashvili (KP) equation. To achieve this goal, we employ the Galilean transformation to derive the dynamic framework associated with the governing equation. Subsequently, we apply the principles of planar dynamical system theory to perform a bifurcation analysis. By incorporating a perturbed element into the established dynamic framework, we explore the potential emergence of chaotic behaviors within the extended KP equation. This investigation is supported by the presentation of phase portraits in both two and three dimensions. Additionally, to ascertain the stability of solutions, we conduct a sensitivity analysis on the dynamic framework employing the Runge–Kutta method. Our results affirm that minor variations in initial conditions have minimal impact on solution stability. Furthermore, employing the modified tanh method, we construct multiple instances of solitons and kinks for the proposed model. [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]

This paper investigates the optical soliton solutions of the Lakshmanan–Porsezian–Daniel equation, a nonlinear evolution equation modeling the propagation of optical soliton waves in materials influenced by parabolic law nonlinearity and spatio-temporal dispersion. Employing both the Kudryashov and new Kudryashov methods, the study captures various soliton waves, deriving both dark and bright soliton solutions. The research delves into the effects of the model parameters on these solutions, presenting the findings through detailed 3D and 2D simulation images. Practical and rapid results achieved using both Kudryashov methods are highlighted, positioning this work as a valuable reference for peers. The study's distinct combination of the Lakshmanan–Porsezian–Daniel equation and the chosen analytical techniques not only introduces novelty but also emphasizes practicality and efficiency. This innovative approach, coupled with its significant contribution to the knowledge base, underscores the research's relevance in the field. [ABSTRACT FROM AUTHOR]

In this paper, we go deeply into the complex Ginzburg–Landau equation with highly dispersive perturbed birefringent fibers having a polynomial law of nonlinearity and acquire three modes of solutions, including solitary wave modes, singular modes, and elliptic function double periodic modes, by using the trial equation method and the complete discrimination system for polynomials. In order to digest the dynamic properties of the model better, we study accurate two-dimensional and three-dimensional images of solutions at specific values. The study of this equation is of great significance for the research and application of superconductors. [ABSTRACT FROM AUTHOR]

In this paper, we use new extended generalized Kudryashov and improved tan (ϑ 2) expansion approaches to investigate Kerr law nonlinearity in the extended Gerdjikov–Ivanov equation in a dense wavelength division multiplexed system. These methods rely on a traveling wave transformation and an auxiliary equation. These approaches successfully extract trigonometric, rational and hyperbolic solutions, along with some appropriate conditions imposed on parameters. To explain the dynamics of soliton profiles, a graphical description of newly discovered solutions is also presented, which exhibits distinct physical significance. The considered methods are recognized as useful and influential tools for creating solitary wave solutions to nonlinear problems in the mathematical sciences. [ABSTRACT FROM AUTHOR]

In this paper, we apply two different methods, namely, the G ′ G -expansion method and the G ′ G 2 -expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation. [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 article studies the Gerdjikov-Ivanov equation with Kerr law nonlinearity in dense wavelength division multiplexed systems and establishes new chirped optical solitons. The approaches used are trial equation method and the complete discriminant system for polynomial method, which are rigorously mathematically derived and calculated, rather than based on certain conditional assumptions. As a result, we obtain nineteen exact solutions and corresponding chirps whose types and quantities are more abundant than existing results, and the outcomes give the equation more profound physical significance. We give the topological stability analysis of the parameters of the solutions. Finally, we draw three-dimensional and two-dimensional graphs of modules of exact solutions and their chirps to verify the existence of solutions and understand their characteristics by taking specific parameters. [ABSTRACT FROM AUTHOR]

In this article, we study for the first time the stochastic Kaup–Newell equation in birefringent fibers. Three integration algorithms are applied, namely, the unified Riccati equation expansion method, the addendum to Kudryashov's method, and the addendum to modified sub‐ODE method. Many types of optical soliton solutions such as Jacobi‐elliptic function solutions, Weierstrass elliptic function solutions, bright solitons, dark solitons, singular solitons, and periodic function solutions are obtained. Numerical schemes give a visual perspective to the solutions derived analytically. [ABSTRACT FROM AUTHOR]

In this work, newly created soliton solutions for ion sound and Langmuir waves with an Atangana–Baleanu fractional (ABF) are given. Symbolic software is used to perform the unified solver method (USM) and Weierstrass elliptic function method (WEFM) in order to solve this model. In terms of hyperbolic functions, extended trigonometric functions, and so forth. Single-wave solutions that were entirely novel and universal were attained. The way the soliton solutions behaved in connection with the two-dimensional (2D) and three-dimensional (3D) graphics was also investigated. The dynamic investigation of the newly created soliton solutions reveals that they have several soliton forms like single-soliton, bell-shaped and mixed-form soliton profiles. This strategy is viewed as promising for handling a range of ABF evolution systems. [ABSTRACT FROM AUTHOR]

This paper tackles the recently introduced (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation (4D-BLMPE) utilized to model wave phenomena in incompressible fluid and fluid mechanics. Modified extended tanh expansion method (METEM) and the new Kudryashov scheme are implemented to produce analytical soliton solutions for the presented equation. The traveling wave transformation is constructed, and the homogeneous balance principle is utilized to apply the two proposed techniques. Furthermore, the flat-kink, smooth-kink, singular, and periodic singular solutions are successfully extracted. Some produced solutions are illustrated graphically to understand the physical meaning of the presented model. Moreover, for the first time in this study, the effect of model parameters on kink soliton dynamics is examined, and graphical representations are depicted and interpreted. [ABSTRACT FROM AUTHOR]

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the P T symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or P T symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process. [ABSTRACT FROM AUTHOR]

This paper addresses Radhakrishnan--Kundu--Lakshmanan equation that arises in the study of soliton dynamics in optical fibers. The bifurcation analysis is carried out and the phase portraits are displayed. The complete discriminant analysis also leads to solitons and other solutions to the model. [ABSTRACT FROM AUTHOR]

In this paper, highly dispersive optical solitons are obtained with the perturbed complex Ginzburg--Landau equation, incorporating the Kerr law of nonlinearity, by the complete discriminant classification approach. A variety of solutions emerge from this scheme that include solitons, periodic solutions and doubly periodic solutions. The numerical sketches support the analytical findings. [ABSTRACT FROM AUTHOR]

The paper presents a description of low-frequency nonlinear dust-acoustic waves in the dusty magnetosphere of Saturn, which contains two types of electrons (hot and cold) following the kappa distribution, magnetospheric ions, and charged dust particles. The Zakharov–Kuznetsov equation for the corresponding conditions is derived, describing the nonlinear dynamics of dust-acoustic waves in the case of low frequencies and a pancake-shaped wave packet along the external magnetic field. It is shown that, under the conditions of Saturn's magnetosphere, solutions of the Zakharov–Kuznetsov equation exist in the form of one-dimensional and three-dimensional solitons. Possible observations of such solitons in future space missions are discussed. [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]

In this study, we will extract new impressive visions for the optical soliton solution and other novel set of the rational soliton solutions to the perturbed Biswas-Milovic equation with Kudryashov's law of refractive index (PBMEWKL) that describes pulses propagation of different categories in optical fiber. Three various schemas are enforcement to extract these new visions. The three proposed techniques that are employed for these purposes are the solitary wave ansatz method (SWAM) the Paul-Painlevé approach method (PPAM) and the Riccati-Bernoulli Sub ODE method (RBSODM). Many new types of soliton solutions have been configured like bright soliton dark soliton combination between bright and dark soliton solutions, W-like shape soliton M-like shape soliton and other rational soliton solutions. The solutions imbedded in this article will play a principal role to understand the exact dynamics of soliton arising in nonlinear optics. The effective of all used algorithms that are examined previously for many other models, usually achieves very good results. Several emblematic 2D and 3D graphs are drawn under specific parameters, which can identify the morphology of optical waves more intuitively obvious. Moreover, we will establish a comparison not only for our achieved results via these three various paths but also with all who studied this model via other techniques. [ABSTRACT FROM AUTHOR]

This paper reveals soliton solutions to ( 4 + 1 )-dimensional Fokas equation, which is an integrable extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations. In wave theory, the Fokas equation plays a crucial role in describing the physical phenomena of waves on the water's surface and beneath it. The observed model is subjected to an extended simple equation method (ESEM) and the Hirota bilinear method (HBM), which disclosed an abundance of soliton solutions in distinct formats, in the form of trigonometric functions, singular, periodic, rational, and exponential solutions. Moreover, we developed a number of solutions, such as the homoclinic breather wave solution,the periodic wave solution, the M-shaped rational wave solution, and the kink with their interaction solution, which are not documented in the literature. Additionally, modulation instability is effectively discussed. Some of the achieved results are explained in 2D, 3D, contour and density graphs. The new results interpreting that these obtained solutions can be a part, to complete the family of solutions and considered methods are effective, simple, and easy to use. [ABSTRACT FROM AUTHOR]

We propose the Hirota bilinearization of the Fokas–Lenells derivative nonlinear Schrödinger equation (FLE) with a non-vanishing background. In the proposed method, we have introduced an auxiliary function to transform the equation into bilinear form. The use of an auxiliary function makes the method simpler than the ones reported earlier. Using the proposed method we have obtained the dark (and bright) one soliton solution. We have also discussed the properties of the obtained soliton and mentioned the criteria upon which the nature of the soliton being dark or bright depends upon. Then we have obtained the dark (and bright) two soliton solution and discussed the respective properties and also through asymptotic analysis showed how the phase between the two individual solitons changes before and after interaction. Eventually we have proposed the scheme for obtaining N soliton solutions. The proposed method can be extended to other nonlinear equations where straightforward bilinearization is not feasible. Later, we have introduced a gauge transformation which transforms the spectral problem of FLE into a spectral problem for the Landau–Lifshitz (LL) spin system. Soliton act as an information carrier and LL system exhibits a variety of nonlinear structures so the study is worth doing. [ABSTRACT FROM AUTHOR]

This paper seeks to introduce the wave structures and dynamics properties of the perturbed Chen–Lee–Liu equation (PCLLE). Using the traveling wave transformation, we derive the corresponding traveling wave system from the original equation and construct a conserved quantity named as Hamiltonian. Subsequently, we establish periodic solutions and the existence of soliton using the bifurcation method. The bifurcation method is a mathematical technique used to study how the qualitative behavior of a system changes as one or more parameters of the system are varied. It involves analyzing the system's equilibrium points and studying how they behave as the parameters are changed. Finally, we construct the exact traveling wave solutions using the complete discriminant system (CDS) of polynomial method (CDSPM) to explicitly validate our findings. [ABSTRACT FROM AUTHOR]

In this paper, the M-lump solutions, the periodic type, and cross-kink wave solutions are acquired. Here, the Hirota bilinear operator is employed. By utilizing the symbolic computation and employing the utilized method, the (3+1)-dimensional Jimbo–Miwa (JM) equation is investigated. Based on the Hirota bilinear form, the soliton solution and periodic wave solution to the mentioned equation, respectively, are obtained. We gained plenty of multiple collisions of lumps. Next, the periodic wave and cross-kink wave have greatly enriched the existing literature on the JM equation. Through the three-dimensional designs, contour design, density design, and two-dimensional design by using Maple, the physical features of these soliton solutions are explained all right. The forms of the attained solutions are one-lump, two-lumps, and three-lumps wave solutions. Then, a class of rogue waves-type solutions to the (3+1)-dimensional JM equation within the frame of the bilinear equation is found. These results can help us better understand interesting physical phenomena and mechanisms. [ABSTRACT FROM AUTHOR]

An extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation is studied in this paper to construct periodic solution, n-soliton solution and folded localized excitation. Firstly, with the help of the Hirota's bilinear method and ansatz, some periodic solutions have been derived. Secondly, taking Burgers equation as an auxiliary function, we have obtained n-soliton solution and n-shock wave. Lastly, we present a new variable separation method for (3+1)-dimensional and higher dimensional models, and use it to derive localized excitation solutions. To be specific, we have constructed various novel structures and discussed the interaction dynamics of folded solitary waves. Compared with the other methods, the variable separation solutions obtained in this paper not only directly give the analytical form of the solution u instead of its potential u y , but also provide us a straightforward approach to construct localized excitation for higher order dimensional nonlinear partial differential equation. [ABSTRACT FROM AUTHOR]

In this research, we investigate the integrability properties of the Schamel–Korteweg–de Vries (S-KdV) equation, which is important for understanding the effect of electron trapping in the nonlinear interaction of ion-acoustic waves. Using the optimal system, we come over reduced ordinary differential equations (ODEs). To deal with reduced ODEs for this problem, Lie symmetry analysis is combined with the modified auxiliary equation (MAE) procedure and the generalized Jacobi elliptic function expansion (JEF) method. The analytical solutions reported here are novel and have a wide range of applications in mathematical physics. [ABSTRACT FROM AUTHOR]

The Fokas-Lenells equation (FLE) including the M-truncated derivative or beta derivative is examined. Using the modified mapping method, new elliptic, hyperbolic, rational, and trigonometric solutions are created. Also, we extend some previous results. Since the FLE has various applications in telecommunication modes, quantum field theory, quantum mechanics, and complex system theory, the solutions produced may be used to interpret a broad variety of important physical process. We present some of 3D and 2D diagrams to illustrate how M-truncated derivative and the beta derivative influence the exact solutions of the FLE. We demonstrate that when the derivative order decreases, the beta derivative pushes the surface to the left, whereas the M-truncated derivative pushes the surface to the right. [ABSTRACT FROM AUTHOR]