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In this study, we explore the positive Gardner Kadomtsov–Petviashivilli (KP) equation using the Generalized Ricatti Equation Mapping Method (GREMM). The KP equation describes the behavior of quasi-one dimensional shallow-water waves in the absence of surface tension and viscosity effects. By employing GREMM, we derive analytical solutions expressed in terms of hyperbolic, rational trigonometric, and exponential functions. These solutions are original and have not been previously documented. They have significant applications in engineering and other applied sciences. By assigning suitable values to the parameters of specific solutions, we generate new graphical structures that enhance our understanding of the physical phenomenon described by this model. The computational effort and resulting outputs demonstrate the effectiveness and power of the proposed technique, which can be extended to other nonlinear models in mathematical physics and various scientific disciplines. [ABSTRACT FROM AUTHOR]

In this work, the (2+1)-dimensional chiral nonlinear Schrödinger equation that describes about quantum field concept in physics and other physical sciences are studied and solved by utilizing the two modern techniques including the polynomial expansion method and the Sardar sub-equation method. We attained different types of soliton solutions that had been applications in different fields of mathematical sciences. The behaviours of attained solutions are periodic, singular and v-shaped soliton solutions. Furthermore, we have investigated the stability of the obtained results. Also, the 3D, 2D, and contour graphics are displayed for the better understanding of the dynamical behaviour of various waves structures extensively. The techniques applied in this article are not used in this model in literature so we say that our findings are new that summarize the novelty of work. The utilize model has applications in physics related phenomenon also obtained results highly valuable in various branches of sciences specially in the transmission of fiber optical. [ABSTRACT FROM AUTHOR]

Nonlinear phenomena, characterized by behaviors that cannot be explained by linear systems and has significant challenges in understanding and modeling. To address this, the mathematical description of such phenomena relies on differential equations. In this study, we investigate the cubic-quintic-septimal nonlinear (7th order nonlinear media) Schrödinger wave equation, which governs the evolution of light beams in a weak non-local medium. The novelty of our study lies in the application of the improved generalized Riccati equation mapping method to obtain exact solutions for the governed equation. This scheme offers a systematic and reliable approach to exploring nonlinear phenomena, contributing to the advancement of nonlinear science and its practical applications. By applying the proposed scheme, a range of exact solutions encompassing trigonometric, rational, exponential, and hyperbolic functions are derived which offer insights into the dynamics of the light beams. Additionally, 2D and 3D graphical illustration are presented to provide a comprehensive demonstration of their dynamical behavior. Furthermore, it is important to highlight the significance of studying the cubic-quintic-septimal nonlinear Schrödinger wave equation which has application in various domains such as quantum mechanics, optics, and nonlinear wave propagation. Understanding of its solutions facilitates the design and optimization of systems involving weak non-local media. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to study various optical soliton profiles of the Schrödinger–Hirota (SH) equation having the parabolic law nonlinearity by utilizing the Sardar-Subequation (SSe) method. First, the general method of the suggested SSe method is presented. Then using traveling wave transformation the suggested equation is converted into the nonlinear ordinary differential equation, where the real and imaginary parts are separated. Furthermore, the proposed method is utilized and novel optical solitons of the suggested SH equation with parabolic law are obtained. Furthermore, the solutions are validated through stability analysis. The exact analytical solutions are graphically simulated where the effects of different parameters on the soliton wave profiles are studied. [ABSTRACT FROM AUTHOR]

In this article, the extended fan method and the positive quadratic function technique are used to solve the (2 + 1) -dimensional Chiral nonlinear Schrödinger equation. Several new outcomes are obtained. Several new results are obtained. We can offer soliton-type solutions, triangular-type solutions, simple and combined solutions using Jacobi's elliptic functions. Using suitable parameters, some graphics are given to observe the evolution of these analytical solutions. Many other nonlinear differential equations can benefit from using the methods proposed in this work. [ABSTRACT FROM AUTHOR]

In this work, various types of soliton solutions for space–time conformable version of (n+1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation were constructed via the approach of modified extended tanh. The derivatives involved were defined to reflect the sense of space–time conformable derivatives. Furthermore, with the help of a fractional wave transformation, the conformable KP equation was reduced into an ODE of a polynomial nature. Mathematica software was used to obtain a system of algebraic equations and then solved. Finally, a graphical illustration for some of the obtained results were provided to show the effect various values of order of the conformable derivative, α (alpha) as well as k, (the sum of linear terms in the equation). [ABSTRACT FROM AUTHOR]

Partial differential equations are frequently employed to depict issues arising across various scientific and engineering domains. Efforts have been made to analytically solve these equations, revealing shortcomings in some widely utilized methods, including modeling deficiencies and intricate solution processes. To address these limitations, diverse analytical methods have been explored. The Ito equation, introduced in 1980, underwent development, leading to the formulation of a fifth-order Ito equation. A seventh-order integrable (3 + 1) -dimensional extended modified Ito equation emerged by augmenting this equation with three additional terms. In this study, novel exact solutions for the equation, absent in existing literature, were derived using the extended hyperbolic function and modified Kudryashov methods. To scrutinize the dynamic behavior of these findings, we presented 3D, contour, and 2D visualizations of select solutions. The results showcase numerous new solutions, underscoring the reliability and efficacy of the employed methods. [ABSTRACT FROM AUTHOR]

With the developing technology, magneto-optical and ferromagnetic materials are gaining importance and are used especially in magneto-optics, ferromagnetism, fluid mechanics, etc. These processes are modeled via Kadomtsev–Petviashvili-type models. In this work, a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system and special cases are considered that simulates electromagnetic, water, and powder-acoustic/ion-acoustic/dust-ion-acoustic waves. As to the novelty of this paper, the travelling wave, soliton solutions of the considered systems are hold by using Bernoulli method which is the well-known ansatz-based method and the analytical method. As far as we know, the obtained solutions are seen for the first time in this study and are important for the development of the use of magneto-optical and ferromagnetic materials in industry and applied sciences, fiber optic communication fields. [ABSTRACT FROM AUTHOR]

The fundamental intention of this article is to find several soliton solutions to the generalized Benjamin–Bona–Mahony equation. The fractional derivatives are described in the conformable sense. These solutions are generated using the functional variable approach. Then, we acquire via this technique some distinctive solutions, such as the singular soliton, periodic soliton, bell-shaped soliton, and kink soliton solutions. The fundamental feature of the demonstrated method is that it's convenient and provides precise, effective solutions to nonlinear equations. The solutions given will offer a thorough examination of this model and any associated occurrences. [ABSTRACT FROM AUTHOR]

Numerous systems and phenomena that appear to fluctuate randomly are mathematically modeled using stochastic processes. In this paper stochastic Phi-4 equation is considered with the multiplicative Wiener process that represents the temporal change of fluctuations. In particle and nuclear physics, the stochastic Phi-4 equation appears to be crucial. Stochastic phenomena represent the microscopic randomness of the nuclear particle. The extended sinh-Gordon expansion approach is used to study the stochastic model in terms of Itô formulation. With the use of this method, variety of solutions from important physical prospectives, such as dark, bright, singular, and periodic solitons has been examined, Furthermore, based on the Itô, the impact of the noise term on Brownian motion is examined. Both 2D and 3D graphs are used to show the impact of the noise term on the obtained soliton solutions. The computations and outcomes demonstrate the method's importance, precision, and effectiveness. Numerous robust nonlinear stochastic differential equations from mathematics, physics, and other relevant domains can be resolved using this method. [ABSTRACT FROM AUTHOR]

In this paper, the technique involving the e x p a function is employed to calculate analytical soliton solutions for three distinct equations: the (3+1)-dimensional mKdV–Zakharov–Kuznetsov equation, the KdV equation, and the (1+1)-dimensional Mikhailov Novikov–Wang integrable equation, which fractional-order in the sense of conformable derivatives. By selecting some parameter values, a diverse spectrum of soliton solutions is obtained, encompassing kink solitons, singular solitons, and periodic-singular solitons. The representation in physical terms enables the examination of authentic multispecies plasmas, plasma models, and frequency ranges. [ABSTRACT FROM AUTHOR]

This paper investigates (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system equations using the improved Sadar subequation method. It uncovers analytical solutions, including dark solitons, bright solitons, and periodic waves. The dynamic behavior of these solutions is illustrated through 2D and 3D plots by adjusting parameters. The results highlight the effectiveness and simplicity of the proposed methods, providing a versatile approach to obtaining various traveling wave solutions. These findings are expected to advance the shallow water theory of ideal fluids. [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]

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 work, exact solutions of the Van der Waals model (vdWm) are investigated with a new algebraic analytical method. The closed-form analysis of the vdW equation arising in the context of the fluidized granular matter is implemented under the effect of time-fractional M-derivative. The vdWm is a challenging problem in the modelling of molecules and materials. Noncovalent Van der Waals or dispersion forces are frequent and have an impact on the structure, dynamics, stability, and function of molecules and materials in biology, chemistry, materials science and physics. The auxiliary equation which is known as a direct analytical method is constructed for the nonlinear fractional equation. The process includes a transformation based on Weierstrass and Jacobi elliptic functions. Wave solutions of the model are analytically verified for the various cases. Then, graphical patterns are presented to show the physical explanation of the model interactions. The achieved solutions will be of high significance in the interaction of quantum-mechanical fluctuations, granular matter and other areas of vdWm applications. [ABSTRACT FROM AUTHOR]

Fractional differential equations are being used to define numerous physical phenomena instead of conventional ordinary or partial differential equations. The secret is to get more generalization of the analysis, hence its solutions. The applications vary between electrical circuit modeling, oscillating circuits, shallow water behavior, viscous fluids, solution transportation, control theory and ultrafast optics. Sine–Gordon Expansion (SGE) Method is being employed hereafter to fully investigate the space–time conformable fractional Vakhnenko–Parkes equation and its modified version. Based on SGE method, the fractional models are transformed into an equivalent ordinary differential equation. The solutions are very important in studying the behavior of ultrafast optics in fibers or waveguides and the wave propagation throughout fiber optics. The solutions are formatted as solitons and are illustrated graphically. The illustrations show that the traveling waves inside fiber optics behave like solitons with traveling peak values according to the wave velocity. One other mode of propagation is to travel in kink shapes. Additionally, the ultrafast optical waves may propagate in shockwaves mode if the waves propagate at ultra-high speeds. [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]

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 the field of nonlinear optics, both fractional and classical-order nonlinear Schrödinger (NS) equations are investigated. However, the fractional-order NS equation has gained widespread acceptance due to its higher compatibility. The space-time fractional nonlinear Schrödinger equation enfolding beta derivative has a wide range of applications in nonlinear optics, quantum computing, Bose-Einstein condensates, wave propagation in complex media, quantum mechanics, and engineering, where understanding wave propagation and nonlinear interactions are diametrical. In this article, the improved Bernoulli sub-equation function (IBSEF) procedure has been used to establish optical soliton solutions in the form of trigonometric, exponential, and hyperbolic functions comprising substantive parameters. These soliton solutions have different shapes, including kink, periodic soliton, singular kink, breathing soliton, and other types. The physical features of the solitons are revealed through three-, two-, contour, and density graphs. The research findings confirm that the IBSEF scheme is effective, straightforward, and applicable for ascertaining soliton solutions in various nonlinear fractional-order models in the fields of physics and communication engineering. [ABSTRACT FROM AUTHOR]

The principal purpose of this research is to utilize the exp-function approach for achieving exact solutions of nonlinear optical fibers, including fractional order in the sense of conformable derivatives. Owing to the algorithm of symbolic computational, Some soliton solutions are obtained, including solitary soliton, singular kink-type soliton, and periodic solitons. These results are generated and developed by using the exp-function. To the greatest of our knowledge, the soliton solutions given in this study could be highly beneficial in comprehending physical phenomena. [ABSTRACT FROM AUTHOR]

Our objective is to find new analytical solutions of the (1 + 1) - and (2 + 1) -dimensional Chiral nonlinear Schrodinger (CNLS) equations using the Jacobi elliptical function method. The CNLS equations play a significant role in the development of quantum mechanics, particularly in the field of quantum Hall effect. Soliton solutions of the considered models are obtained such as, cnoidal solutions, the hyperbolic solutions and the trigonometric solutions. The obtained analytical solutions are new in the literature. The stability conditions of these solutions are also given. The obtained stable solutions are presented graphically for some specific parameters. Moreover, the conditions of modulational instability for both models are provided. The proposed method can be useful to obtain the analytical solutions of nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In this work, we investigate diverse analytical solutions and modulation instability of the nonlinear Schrödinger equation with an anti-cubic nonlinear term. We use the traveling wave transformation and the New Generalized Extended Direct Algebraic Method to perform a variety of exact traveling wave soliton-like solutions. The obtained solutions are among other trigonometric function solutions, complex soliton solutions, and rational solutions. A particular behavior has been depicted in self-focusing and defocusing nonlinearity where dark soliton changes to super rogue waves and breather soliton is obtained. We then use the linearizing scheme to get the growth rate of modulation instability. For a specific value of the excitation wave number and a specific time of simulation, diverse waveforms of the modulated waves are obtained. We have demonstrated that the combined dispersion and nonlinear terms, as well as the anti-cubic nonlinear term, can develop modulation instability. To confirm the predictions based on the analytical results, we use the direct numerical simulation, from which we have displayed a modulation wave of the growth rate modulation instability. It results equally that the anti-cubic nonlinearity is an important object to control the propagation of the solitonic waves in the nonlinear systems. [ABSTRACT FROM AUTHOR]

This article will investigate the exact solitary wave solutions of the (3 + 1) generalized nonlinear wave equation with gas bubbles. Gas bubble-containing liquids are frequently seen in physics, engineering, science, and other fields. We explored this model using the extended auxiliary equation method. This method gives many essential solitary wave solutions, ranging from exponential to trigonometric and hyperbolic wave solutions. The investigated solutions are entirely different and new to the reported model. These demonstrated the usefulness and efficiency of the applied method for finding partial differential equations' solitary wave solutions. Additionally, utilizing Maple mathematical software in both 2D and 3D as well as the contour plots, some of the calculated solutions are plotted to comprehend the physical structures of the investigated model. [ABSTRACT FROM AUTHOR]

In mathematics, physics, and engineering, establishing numerical or analytical solutions for fractional mathematical models for specific phenomena and developing fractional mathematical models for specific phenomena are important topics. In this work the (G ′ G) - expansion method with the generalized fractional derivative has been used to obtained the explicit solutions of nonlinear fractional Tzitz e ´ ica type nonlinear evolution equations. These models are tested to illustrate the pertinent features of the proposed method. [ABSTRACT FROM AUTHOR]

Searching for soliton solutions of nonlinear partial differential equations is one of the most interesting and important areas of science in the field of nonlinear phenomena. Soliton is a localized wave with exponential wings or is a localized wave with an infinite support. In this work, we study two extensions of (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Based on the simplified Hirota's method and the Cole-Hopf transformation method, new multiple front wave solutions are obtained for both versions. [ABSTRACT FROM AUTHOR]

In this study, a robust radial basis function neural network (RBF-NN) is developed for predicting Iranian crude oil viscosity in an extensive and precise way. Experimental data incorporate the PVT data of 720 samples gathered from Iranian central, southern, and offshore oil fields. The proposed RBF-NN model uses temperature, pressure, and parameters obtained by PVT analyses including oil and gas specific gravity, and solution gas–oil ratio as independent variables. The evaluation process was employed in three different regions; above, at, and below the bubble point pressure (Pb). The proposed RBF-NN model outputs were evaluated statistically using experimental data, and the results were compared side by side with the results of the previous studies including a multilayer perceptron neural network (MLP-NN) and the five well-established semiempirical equations. Sensitivity analyses were performed using the numeric sensitivity analyses (NSA) method and the results showed the highest and lowest impacts on the predicted crude oil viscosity between input parameters related to oil and gas specific gravity with 42 and 0%, respectively. The results show the proposed RBF-NN model with the average absolute relative deviation (AARD%) of 1.69, 4.56, and 2.04% for above, at, and below the bubble point pressure (Pb) regions, respectively, is the most precise and consistent method for predicting crude oil viscosity compared with those published in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we studied a nonlinear wave equation that models the propagation of optical solutions in a weakly nonlocal and parabolic competing nonlinear medium. The exact traveling wave type solutions formulated in hyperbolic functions, rational and trigonometric functions multiplied by some exponential functions to the governing equation are determined explicitly by the extended form of the Kudryashov method. We have examined the behavior of the modulation instability (MI) growth rate. To substantiate the stability of the obtained dark and bell-shaped solitons, we use the split-step Fourier method. In addition, the conservation laws describing significant physical concepts of this equation are examined. Compared to the obtained results with Younis et al. (J Nonlinear Opt Phys Mater 24(04):1550049, 2015), Zhou et al. (Optik 124(22):5683–5686, 2013; Proc Rom Acad Ser A 16(2):152–159, 2015) and Akinyemi et al. (Optik 230:166281, 2021), we have shown the propagation of the solitonic waves which sometime tilt from right to left with stable shape. [ABSTRACT FROM AUTHOR]

The major purpose of this study is to seek diverse soliton solutions to the nonlinear discretized mKdv lattice system including fractional-order in the sense of conformable derivative. The rational forms of trigonometric, hyperbolic, and Jacobi elliptic functions are employed to generate these exact solutions. The auxiliary rational structure is used for obtaining exact solutions which provide solitary, periodic, and soliton varieties of traveling wave solutions. The solutions obtained will provide a comprehensive study of this model and any related events. [ABSTRACT FROM AUTHOR]

The ultimate focus of this research is to provide soliton solutions to the nonlinear differential-difference equations via conformable fractional derivatives which arise from the polarization of the ferroelectric nanoparticles. The investigation of achieved soliton solutions will provide effective results of ferroelectric materials and their related phenomena. [ABSTRACT FROM AUTHOR]

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

The purpose of this work is to seek various innovative exact solutions using the new Kudryashov approach to the nonlinear partial differential equations (NLPDEs). This technique obtains novel exact solutions of soliton types. Moreover, several 3D and 2D plots of the higher dimensional Klein-Gordon, Kadomtsev-Petviashvili, and Boussinesq equations are demonstrated by considering the relevant values of the aforementioned parameters to exhibit the nonlinear wave structures more adequately. The new Kudryashov technique is an effective, and simple technique that provides new generalized solitonic wave profiles. It is anticipated that these novel solutions will enable a thorough understanding of the development and dynamic nature of such models. [ABSTRACT FROM AUTHOR]

A new fractional-order derivative operator is presented and applied to nonlinear Sasa-Satsuma equation having Kerr nonlinearity law with third-order dispersion. Two novel families of singular, dark and bright soliton solutions for the fractional-order nonlinear Sasa-Satsuma equation were obtained by considering the modified simple equation method and the F-expansion method. Behavior of these soliton solutions are illustrated in 3d graphs, contour plots and 2d plots. Furthermore, periodic singular solutions of the equation are obtained by both analytical techniques. Optical type soliton solutions of the nonlinear Schrödinger equation is generated as well. To show out the behavior of the modulation gain spectra with the effects of the fractional derivative order, the linear stability technic was used. The effects of the fractional derivative order have been investigated in normal and anomalous dispersion regime associated to the Kerr nonlinearity effects. MI gain spectra were depicted by choosing adequate parameters of the fractional nonlinear Sasa-Satsuma equation. [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]

In this article, based on the concepts of the extended tanh expansion method, the general nonlinear magneto-optic waveguides that maintain parabolic-nonlocal law of refractive index is discussed. Then, the physical properties and structures of these obtained solutions are analyzed by means of some graphics. The obtained results can be used in describing the substantial understanding of the studious structures as well as other related non-linear physical structures. [ABSTRACT FROM AUTHOR]

The interfacial dynamics of a system are significant in understanding the system's behavior. One of the special models of these systems which naturally arises in material sciences is the phase separation (or spinodal decomposition) of binary alloys. A nonlinear evolution equation known as the Cahn–Hilliard equation is used to establish the mathematical modeling of a binary alloy for phase separation. Apart from trivial solutions, the Cahn–Hilliard equation is a fourth-order nonlinear equation without an analytical solution. This study investigates a second-order splitting finite difference scheme based on the 2D Crank–Nicolson technique to approximate the solution of the 2D Cahn–Hilliard problem with Neumann boundary conditions. Using the inherent error estimation of the 2D Crank–Nicolson method, we have shown that the proposed scheme is second-order accuracy and we have proved that the scheme has a unique solution. In addition, we demonstrated that the suggested technique retains mass conservation while decreasing total energy. Furthermore, we chose two numerical examples, one with a special initial value and the other with a random initial value, to confirm the accuracy of the proposed approach. These numerical experiments display physical features of the Cahn–Hilliard model such as coarsening dynamics and spinodal decomposition obtained by experimental studies of binary alloys. [ABSTRACT FROM AUTHOR]

In this article, we introduce analytical-approximate solutions of time-fractional generalized Pochhammer-Chree equations for wave propagation of elastic rod by means of the q-homotopy analysis of the transform method (q-HATM). In the Caputo sense, basic concepts for fractional derivatives are defined. Several examples are given and the results are illustrated via some surface plots to present the physical representation. The results show that the current methodology is productive, powerful, efficient, easy to use, and ready to incorporate a wide variety of partial fractional differential equations. [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]

A special type of equation which have variable coefficients and contains terms with higher order nonlinearity is studied. We give conditions under which exact solutions for it can be obtained using a special computational method. From the solutions presented here, novel traveling wave solutions for the Chen-Lee-Liu (CLLE) are presented. Finally, we give some conclusions. [ABSTRACT FROM AUTHOR]

In this work, a variety of optical soliton solutions are derived for a nonlinear generalized equation with variable coefficients. At first, a computational approach is used to obtain solutions for the proposed model for a particular case. After, a generalized approach is considered to obtain other type of solutions given in a more general form. From the model considered here, the classical perturbed Fokas-Lenells equation is obtained and new optical soliton solutions for this last case are presented. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]

In this article we have acquired exact solitary wave solutions for the truncated Ω - fractional Lakshamanan–Porsezian–Daniel model with Kerr, parabolic, and anti-cubic nonlinear laws employing extended auxiliary technique. Diverse set of exponential function solutions acquired relying on a map between the considered equation and an auxiliary ODE. Obtained solutions are recast in several hyperbolic and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solution. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model using mathematical software. In the end, the modulation instability analysis of the proposed model with normal derivatives is also carried out for Kerr, parabolic, and anti-cubic nonlinear laws. For these three cases, dispersion relations are obtained and explained with plots. Results turned out here may be useful in network technology to study the characteristics of fiber optic communication over inter-continental distances. [ABSTRACT FROM AUTHOR]

The propagation of optical solitons via nonlinear metamaterials with cubic-quintic nonlinearity, detuning intermodal dispersion, self steepening effect, and nonlinear third and fourth-order dispersions is the focus of this study. To find the optical solitons and other solutions, the extended sinh-Gordon equation expansion method is applied to the aforementioned model. As a result, dark, bright, combined dark–bright, singular, combined singular soliton, and singular periodic wave solutions are obtained. To our best knowledge, the application of the method to the model, and the acquired combined soliton solutions are novel. To understand the nonlinear propagation theory of solitons in metamaterials, the reported outcomes can be enriched by the soliton theory. [ABSTRACT FROM AUTHOR]

This paper applies function transformation method to obtain under certain conditions bright, dark, kink and W-shaped dark solitons waves solutions to the modified Complex Ginzburg Landau Equation.The relevant aspect that is encountered using the applied method is the ability to lead to many types of interesting soliton solutions to the model. These new obtained solutions can be useful in many applications such as communication, medicine, hydrodynamic, thermodynamic just to name a few and can allow to explain physical phenomena observed in fundamental sciences and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we implemented extended e x p - φ ξ -expansion method for some exact solutions of (3 + 1)-dimensional nonlinear Schrödinger equation (NLSE) and coupled nonlinear Schrodinger’s equation. The solutions we obtained are hyperbolic, trigonometric and exponential solutions. We observed that these solutions provided the equations through Mathematica 11.2. Apart from that, we have shown the graphics performance of some of the solutions found. This method has been used recently to obtain exact traveling wave solutions of nonlinear partial differential equations. The results achieved in this study have been confirmed with computational software Maple or Mathematica by placing them back into NLFPDEs and found them correct. We posited that the approach is updated to be more pragmatic, efficacious, and credible and that we pursue more generalized precise solutions for traveling waves, like the solitary wave solutions. [ABSTRACT FROM AUTHOR]

Based on two algorithm integrations, such as the exp (- Φ (ξ)) -expansion method and the hyperbolic function method, we build dark, bright and trigonometric function solution to the Klein–Gordon equations with M-fractional derivative of order α . By adopting the travelling-wave transformation, the constraint condition between the model coefficients and the travelling-wave frequency coefficient for the existence of soliton solutions is also obtained. Moreover, miscellaneous soliton solutions obtained is depicted in 3D and 2D. [ABSTRACT FROM AUTHOR]

In this work, we study the exact traveling wave solutions of (2 + 1) -dimensional Chiral nonlinear Schrödinger equation with the aid of generalized auxiliary equation method. The aforementioned model is used as a governing equation to discuss the wave in quantum field theory. The suggested technique is direct, effective, powerful, and offers constraint conditions to ensure the existence of solutions. The solutions obtained are bright solitons, dark solitons, singular solitons, mixed solitons, periodic waves, exponential, rational, and complex solutions that are relevant in various applications of applied science. Finally, some solutions are depicted in two and three dimensional to better understand the behavior of the considered model. [ABSTRACT FROM AUTHOR]

The fourth-order nonlinear Boussinesq water wave equation, which explains the propagation of long waves in shallow water, is explored in this article. We used the Lie symmetry approach to analyze the Lie symmetries and vector fields. Then, by using similarity variables, we obtained the symmetry reductions and soliton wave solutions. In addition, the Kudryashov method and its modification are used to explore the bright and singular solitons while the Hirota bilinear method is effectively used to obtain a form of breather and lump wave solutions. The physical explanation of the extracted solutions was shown with the free choice of different parameters by depicting some 2-D, 3-D, and their corresponding contour plots. [ABSTRACT FROM AUTHOR]

In this paper, we find new soliton solutions of the Sasa-Satsuma equation according to the Sardar sub-equation scheme. Different forms of soliton solutions, including the optical dark, optical bright, and optical singular function solutions are formally extracted. To better realize the physical phenomena of the gained solutions, some physical explanations of the exhibited solutions under suitable parameters for the physical values via the contour and 3D simulations are given. The confrontation between the dispersion and nonlinear terms has permitted survey the treatment of the Modulation Instability gain spectra. [ABSTRACT FROM AUTHOR]

A new local fractional-order derivative operator is introduced and the Lakshmanan–Porsezian–Daniel (LPD) model is interpreted via this operator. New analytical solutions to the LPD equation is presented by Jacobi elliptic functions and an anzätz method. The complex-valued LPD equation includes a nonlinear term which is considered from three different cases: Kerr, parabolic and anti-cubic law of nonlinearities. For each case, dark, bright, singular optical soliton solutions related with optical fibers are presented. Simulations representing behavior of these solutions for different parameter values are provided. [ABSTRACT FROM AUTHOR]

This article examines the complex Ginzburg–Landau equation with the beta time derivative and analyze its optical solitons and other solutions in the appearance of a detuning factor in non-linear optics. The kink, bright, W-shaped bright, and dark solitons solution of this model are acquired using the modified Exp-function and Kudryshov methods. The model is examined with quadratic-cubic law, Kerr law, and parabolic laws non-linear fibers. These solitons emerge with restrictive conditions that ensure their existence are also presented. Furthermore, the obtained and precise solutions are graphically displayed, illustrating the impact of non-linearity. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. [ABSTRACT FROM AUTHOR]