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In the present paper, extensive research is conducted on a 2D generalized Kadomtsev–Petviashvili (2D-gKP) equation which models water waves with long wavelengths. The study starts by establishing Hirota D -operator forms of the governing equation using the Bell polynomial method (BPM). Through checking the three-soliton condition for the 2D-gKP equation, its integrability is then examined, and as a consequence, single-, double-, and (special) triple-soliton waves are derived from the modified Hirota method. In the end, after deriving lump and breather waves of the 2D-gKP equation through ansatz methods, a theoretical discussion along with its proof is presented. Attempts have been made to elucidate the physical features of nonlinear waves by displaying such waves in 2D and 3D postures. The paper's findings contribute certainly to research on nonlinear waves of KP-type equations. [ABSTRACT FROM AUTHOR]

This study aims to investigate a financial system consisting of four ordinary differential equations associated with the rate of interest, investment demand, price index, and the density of financial information gained by the population. The equilibrium and local stability of the system are investigated numerically. The impact of saving amounts and the rate of investment demand increases after getting financial information on the system are discussed. The findings of the study are verified graphically. It is found that the system becomes stable if the rate of investment demand increases after getting financial information kept at a certain level, such that the savings amount is maintained at a higher level. Also, the bifurcation diagrams of the system for various significant parameters that affect the system's stability have been depicted. [ABSTRACT FROM AUTHOR]

Viral infections pose significant threats to public health globally. Understanding the behavior, transmission, and epidemiology of viruses is essential for developing strategies to prevent, control, and manage outbreaks. Mathematical models help in identifying emerging viral pathogens, assessing their risks, and implementing effective public health measures to mitigate their impact. In this work, we formulate the dynamics of Covid-19 viral infection with the effect of vaccination in fractional framework. Our study is mainly concerned with the dynamical behavior and qualitative analysis of Covid-19 dynamics. The model is investigated for basic properties and the threshold of the system is determined. To scrutinize the solution of the recommend system, we use the fixed point theorems of Schaefer and Banach to evaluate the existence and uniqueness of solutions. Moreover, sufficient conditions for the Ulam–Hyers stability of system is established through mathematical skills. We examine the solution pathways of our model through a numerical scheme to show the importance different input parameters of the system. Our findings emphasize the pivotal role of asymptomatic carriers and losing rate of immunity as critical determinants that can heighten the challenge of controlling Covid-19. Vaccination rate and the fractional parameters are attractive parameters while asymptotic fraction poses a significant risk since it has the ability to spread the illness to uninfected areas. We suggest that effective control of the transmission rate can effectively regulate the intensity of Covid-19 transmission. [ABSTRACT FROM AUTHOR]

The (4+1)-dimensional Davey-Stewartson-Kadomtsev-Petviashvili equation is explored in the present work, revealing its complex dynamics and solitary wave solutions. Modeling ocean and tidal waves, particularly tsunami and long water waves, depends significantly on this nonlinear equation. Additionally, these models can be used to simulate internal and external waves in rivers and oceans as well as wave packets in water with a finite depth. The Sardar subequation method, new Kudryashov's method, and (1 ϑ (ζ) , ϑ ′ (ζ) ϑ (ζ)) method are investigated to discover novel solitary wave solutions in the terms of hyperbolic, trigonometric and rational functions. A wide variety of solitons, as dark, bright, periodic, singular, combined dark-singular solitons and, combined dark-bright are obtained by these techniques. By taking accurate parameter values, certain three-dimensional and two-dimensional graphs are plotted to improve the physical description of solutions. The intriguing field of nonlinear waves and dynamic systems is signaled to readers by this work, which suggests a major advancement in understanding the intricate and unexpected behavior of this model. [ABSTRACT FROM AUTHOR]

In this work, a new adaptive numerical method is proposed for solving nonlinear, singular, and stiff initial value problems often encountered in real life. Starting with a fixed step size, the new method's performance can be significantly enhanced by introducing an adaptive step-size approach. The qualitative properties of the proposed method have been investigated to determine the efficiency and reliability of the method. The proposed method is of fifth-order accuracy, zero stable, L -stable, and consistent. In addition, the proposed method is convergent, and its stability properties are also shown through its Order Stars. Finally, numerical experiments are conducted to illustrate the performance of the method. The results obtained show that the proposed method compares favourably with existing methods. [ABSTRACT FROM AUTHOR]

In this analysis, we use the high order cubic B-spline method to create approximating polynomial solutions for fractional Painlevé and Bagley-Torvik equations in the Captuo, Caputo-Fabrizio, and conformable fractional sense concerning boundary set conditions. Using a piecewise spline of a 3rd-degree polynomial; the discretization of the utilized fractional model problems is gained. Taking advantage of the Taylor series expansion; the error order behavior spline theorem is proved. We demonstrate applications of our spline method to several certain kinds including the 1st(2nd) Painlevé and Bagley-Torvik fractional models. For more detail, using Mathematica 11 several drawings and many tables were calculated and their explanations were mentioned. The computational results indicate that the suggested spline approach is most acceptable in terms of cost efficiency and precision of calculations. Highlight, conclusion, and future notes are provided to extract the ability of the discussed approach and the tendency of the utilized fractional models to extrapolate new application areas in the meshless numerical training. [ABSTRACT FROM AUTHOR]

A lot of numerical formulations of physical phenomena contain 9 th -order BVPs. The presented probe intends to consider the spline solutions of the 9 th -order boundary value problems using Cubic B Spline(CBS). Ninth order boundary value problems arise in the study of laminar viscous flow in a semi-porous channel, astrophysics, hydrodynamic & hydro-magnetic stability. The derived technique is exceptionally useful and is appropriate for such kinds of linear and non-linear boundary value problems. Cubic B Spline(CBS) is successfully applied to mathematical models. The outcomes are contrasted to those presented in the literature, revealing that the introduced technique leads to an advisable estimation of the exact solution. [ABSTRACT FROM AUTHOR]

In this article, a new iterative method of the rational type having fifth-order of accuracy is proposed to solve initial value problems. The method is self-starting, stable, consistent, and convergent, whereas local truncation error analysis has also been discussed. Furthermore, the method has been analyzed with a variable stepsize approach that increases performance while taking fewer steps with acceptable local errors. The method is also tested against some existing fifth-order methods having rational structure. The proposed one outperforms concerning maximum absolute error, final absolute error, average error, and norm, while CPU time computed in seconds is comparable. Furthermore, stiff, singular, and singularly perturbed problems for single and system of differential equations chosen for simulations yielded minor errors when solved with the new rational method. [ABSTRACT FROM AUTHOR]

The purpose of this research is to study, investigate, and analyze a class of temporal time-FNEE models with time-FCDs that are indispensable in numerous nonlinear wave propagation phenomena. For this purpose, an efficient semi-analytical algorithm is developed and designed in view of the residual error terms for solving a class of fifth-order time-FCKdVEs. The analytical solutions of a dynamic wavefunction of the fractional Ito, Sawada-Kotera, Lax's Korteweg-de Vries, Caudrey-Dodd-Gibbon, and Kaup-Kupershmidt equations are provided in the form of a convergent conformable time-fractional series. The related consequences are discussed both theoretically as well as numerically considering the conformable sense. In this direction, convergence analysis and error estimates of the developed algorithm are studied and analyzed as well. Concerning the considered models, specific unidirectional physical experiments are given in a finite compact regime to confirm the theoretical aspects and to demonstrate the superiority of the novel algorithm compared to the other existing numerical methods. Moreover, some representative results are presented in two- and three-dimensional graphs, whilst dynamic behaviors of fractional parameters are reported for several α values. From the practical viewpoint, the archived simulations and consequences justify that the iterative algorithm is a straightforward and appropriate tool with computational efficiency for long-wavelength solutions of nonlinear time-FPDEs in physical phenomena. [ABSTRACT FROM AUTHOR]

The ongoing study deals with various forms of solutions for the biological population model with a novel beta-time derivative operators. This model is very conducive to explain the enlargement of viruses, parasites and diseases. This configuration of the aforesaid classical scheme is scouted for its new solutions especially in soliton shape via two of the well known analytical strategies, namely: the extended Sinh-Gordon equation expansion method (EShGEEM) and the Exp a function method. These soliton solutions suggest that these methods have widened the scope for generating solitary waves and other solutions of fractional differential equations. Different types of soliton solutions will be gained such as dark, bright and singular solitons solutions with certain conditions. Furthermore, the obtained results can also be used in describing the biological population model in some better way. The numerical solution for the model is obtained using the finite difference method. The numerical simulations of some selected results are also given through their physical explanations. To the best of our knowledge, No previous literature discussed this model through the application of the EShGEEM and the Exp a function method and supported their new obtained results by numerical analysis. [ABSTRACT FROM AUTHOR]

This article presents soliton solutions to a generalized nonlinear Fokas-Lenells equation via the Sine-Gordon expansion method. To uncover the clear picture of the gained solutions, the two and three-dimensional figures for the solutions are given. It is shown that the proposed methodology provides powerful mathematical tools for obtaining the exact traveling wave solutions of different nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

• New solitons solutions for the CGL equation with Kerr law nonlinearity were constructed. • The CGL equation is considered in classical, truncated M -fractional derivative, beta fractional derivative, and conformable fractional derivative types. • The solutions of these models were found using the Unified technique. • The Physical meaning for the obtained solutions is graphically investigated. The study of optical soliton solutions plays a vital role in nonlinear optics. The foremost area of optical solitons research encompasses around optical fiber, telecommunication, meta -surfaces and others related technologies. The aim of this work is to integrate optical soliton solutions of the complex Ginzburg-Landau (CGL) model with Kerr law nonlinearity, also showing the effect of diverse fraction derivative and comparing it with the classical form. Here, the local derivative is used as the conformable wisdom known as the truncated M -fractional derivative, beta fractional derivative, and conformable fraction derivative. We also deliberated on some assets satisfied by the derivative. The CGL model is useful to describe the light propagation in optical communications, optical transmission, and nonlinear optical fiber. Under the right circumstances, the affectionate unified scheme is implemented for the complex Ginzburg-Landau model to generate the optical wave pattern. For α 1 = 2 α 2 , the unified scheme generates the solution of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme provides some novel optical solitons such as periodic waves, periodic with rogue waves, breather waves, different types of periodic rogue waves, a singular soliton solution, and rogue with the periodic wave for the special value of the free parameters. For α 2 = - ρ + 6 α 1 8 , the unified scheme generates the solutions of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme offers some fresh optical solitons such as periodic rogue waves, multi-rogue waves, double periodic waves, and periodic waves. In numerical argument, the wave patterns are offered with 3-D and density plots. To test the stability of the obtained solutions, we show diverse fractional forms such as beta time fractional, and conformable time fractional derivative and compare these fractional derivatives with their classical form in 2D plots. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of the obtained solutions, emphasizing their distinct features and depictions using unified technique. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations. [ABSTRACT FROM AUTHOR]

The primary objective of this study aims to carry out a more thorough investigation into a fractional nonlinear double dispersive equation that is used to represent wave propagation in an elastic, inhomogeneous Murnaghan's rod. By Murnaghan's rod, we mean the materials, which include the constitutive constant, Poisson ratio, and Lame'́s coefficient, are considered to be compressible in nature forming up the elastic rod. To solve the fractional version of Murnaghan's rod problem, we employed β -fractional and M -Truncated fractional derivative. Regarding the extraction of polynomial and rational function solutions of the Murnaghan's rod problem, which degenerate into several wave solutions including solitary, soliton (dromions), as well as periodic wave solutions. We employ the well-known unified and new auxiliary equation methods of nonlinear sciences. A finite series of certain functions satisfying an ordinary differential equation of first order, second degree is used to represent the projected solution. Based on the given approach, numerous types of solutions for exponential, hyperbolic, and trigonometric functions are generated. In this research study, the behavior of a dynamical planer system has been examined by giving various values to parameters and by depicting every possible situation as a phase portrait. The sensitivity analysis, where the soliton wave velocity and wave number parameters influence the water wave singularity, is demonstrated using the wave profiles of the constructed dynamical structural system. With the use of graphs, we have simulated the solitons to determine their kinds. All of solutions found in this manuscript is been confirmed through back substituting them into the original model using computational software. • An investigation to the wave propagation in an elastic, inhomogeneous Murnaghan's rod is considered. • The solutions under Beta and M-Truncated fractional derivatives for this model were found by two different analytical techniques. • The behavior of the dynamical planer system has been examined by depicting every possible situation as a phase portrait. • The sensitivity analysis is demonstrated using the wave profiles of the constructed dynamical structural system. [ABSTRACT FROM AUTHOR]

The objective of this research is to investigate the nonlinear Landau–Ginzburg–Higgs equation, which characterizes nonlinear solitary waves exhibiting distant and feeble scattering interactions among tropical tropospheres and mid-latitudes. Additionally, the study will examine the interchange of mid-latitude Rossby waves and equatorial waves within this context. In this research article, we focus on obtaining exact traveling wave solutions for the Landau–Ginzburg–Higgs equation using a new extended direct algebraic technique. The obtained soliton solutions include various types such as combined and multiple bright-dark, periodic, bright, and multiple bright-periodic. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations allow us to visualize the behavior of the wave velocity and wave number as the parameters change. Additionally, we conduct a chaotic analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form traveling wave solitary solutions for a wide range of nonlinear evolution equations. • Solitary wave solutions to the LGH equation are derived. • The performance is investigated via a new extended direct algebraic technique. • The dynamical behavior of the LGH equation is studied by chaotic analysis. [ABSTRACT FROM AUTHOR]

Herein, the extended coupled Kadomtsev–Petviashvili equation (CKPE) with lateral dispersion is investigated for studying the atmospheric blocking in two layers. A variety of new types of polynomial solutions for the CKPE is obtained using the unified method. Furthermore, we use the Hamiltonian systems with two degrees of freedom to discuss the stability of the obtained solutions through the bifurcation diagrams. [ABSTRACT FROM AUTHOR]

In this article, the nonlinear fractional Kudryashov's equation and the space–time fractional nonlinear Tzitzeica–Dodd–Bullough (TDB) equation are solved using the new auxiliary equation method, which yields innovative analytical solutions using β and M -Truncated fractional derivatives. The fractional wave and Painlevé transformations are implemented to transform the space and time fractional nonlinear equations into a nonlinear ordinary differential equation. The model solutions are compared by utilizing the two fractional derivatives. Hyperbolic, trigonometric, rational, exponential, and other sorts of soliton solutions are discovered, and these forms of the outcomes illustrate the superiority of the method's uniqueness. The solutions provided in this piece are sophisticated and comprehensive, and the results of the literature review are one instance of the findings. The key advantage of this approach over remedies is that it possesses higher options with flexible constraints. By plotting 3D as well as 2D graphs of the acquired results and analyzing the effect of the fractional parameter ρ on the wave profiles of the phenomenon, it has been determined that the fractional parameter significantly affects the wave profiles. The findings are carried out in such a way as to showcase the applicability and expertise of fractional derivatives and the proposed approach to evaluate several nonlinear fractional partial differential equations. Finally, the comprehensive sensitivity analysis of the proposed models is done by first transforming it into the format of a planer dynamical system using the Galilean transformation. • New analytical solutions for two different fractional nonlinear models were constructed. • The solutions of these models were found using the new auxiliary equation method. • The Physical meaning for the obtained solutions is graphically investigated. [ABSTRACT FROM AUTHOR]

• New solitons solutions for two different nonlinear models were constructed. • The solutions of these models were found using the rational (G'/G,1/G)-expansion technique. • The Physical meaning for the obtained solutions is graphically investigated. In the electrical transmission lines, the processing of cable signals distribution, computer networks, high-speed computer databases and discrete networks can be investigated by the modified Zakharov-Kuznetsov (mZK) equation as a data link propagation control model in the study of nonlinear Schrödinger type equations as well as in the analysis of the generalized stationary Gardner equation. The proposed Gerdjikov–Ivanov model can be used in the field of nonlinear optics, weakly nonlinear dispersion water waves, quantum field theory etc. In this work, we developed complete traveling wave solutions with specific t-type, kink type, bell-type, singular solutions, and periodic singular solutions to the proposed mZK equation and the Gerdjikov-Ivanov equation with the aid of the double G ′ / G , 1 / G - expansion method. These settled solutions are very reliable, durable, and authentic which can measure the fluid velocity and fluid density in the electrically conductive fluid and be able to analysis of the flow of current and voltage of long-distance electrical transmission lines too. These traveling wave solutions are available in a closed format and make them easy to use. The proposed method is consistent with the abstraction of traveling wave solutions. [ABSTRACT FROM AUTHOR]

• New solitons solutions for two different nonlinear models were constructed. • The solutions of these models were found using the rational (G'/G)-expansion technique. • The Physical meaning for the obtained solutions are graphically investigated. For uni-directional wave transmission in the smooth bottom of shallow sea water and the superconductivity of nonlinear media with dispersion systems, the (1 + 1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations are of particular interest in research to the academics. Analytical wave solutions to the stated models have been successfully constructed in this study, which might have considerable implications in describing the nonlinear dynamical behavior associated with the phenomena. The models we aim to uncover have been put into the form of differential equations with one characteristic variable expending the wave transformation coordinate and, thereupon, the rational (G ′ / G) -expansion technique is executed. Using the considered technique, diverse soliton solutions in suitable forms arrayed to trigonometric, rational, and hyperbolic functions have been determined. The achieved solutions are figured out in 3D profiles, assigning the free parameters involved in solutions to particular values and discussed their physical significance to bring out the inner context of the tangible incidents in the natural domain. The rational (G ′ / G) -expansion approach is efficient, concise, and capable of finding analytical solutions to other nonlinear models that can be considered in subsequent studies. [ABSTRACT FROM AUTHOR]

• New solitons solutions for the NLS and MTs models were constructed. • The performance is discussed through the generalized Kudryashov method. • The Physical meaning for some of the obtained solutions are graphically investigated. The perturbed nonlinear Schrödinger (NLS) equation and the nonlinear radial dislocations model in microtubules (MTs) are the underlying frameworks to simulate the dynamic features of solitons in optical fibers and the functional aspects of microtubule dynamics. The generalized Kudryashov method is used in this article to extract stable, generic, and wide-ranging soliton solutions, comprising hyperbolic, exponential, trigonometric, and some other functions, and retrieve diverse known soliton structures by balancing the effects of nonlinearity and dispersion. It is established by analysis and graphs that changing the included parameters changes the waveform behavior, which is largely controlled by nonlinearity and dispersion effects. The impact of the other parameters on the wave profile, such as wave speed, wavenumber, etc., has also been covered. The results obtained demonstrate the reliability, efficiency, and capability of the implemented technique to determine wide-spectral stable soliton solutions to nonlinear evolution equations emerging in various branches of scientific, technological, and engineering domains. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Drinfel'd–Sokolov–Wilson equation (DSWE) and Boiti Leon Pempinelli equation (BLPE) in the conformable sense. The sine–cosine method is utilized to achieve various traveling wave solutions to the suggested nonlinear systems. It is an easy approach to use and does not require sophisticated mathematical software or a knowledgeable coder. It can also be used for various linear and nonlinear fractional issues, making it pervasive. The obtained solutions in the form of solitons emerge with the necessary constraints to ensure their existence. The obtained results hold significant role in elucidating some important nonlinear problems in applied sciences and engineering. • The DSWE and BLPE in the conformable sense are considered. • The sine–cosine method is utilized to achieve various solutions to the suggested models. • The obtained soliton solutions emerge with the necessary constraints to ensure their existence. • Physical explanations are discussed for the obtained solutions. [ABSTRACT FROM AUTHOR]

The present study is made to develop the fractional model of non-Newtonian Casson and Williamson boundary layer flow in the fluid flow taking into account the heat flux and the slip velocity. The temperature and the velocity fields, of the steady boundary layer flow, are generated by a stretched sheet with a non-uniform thickness. The governing non-linear system of PDEs is transformed into a non-linear set of coupled ODEs and then solved by using the Vieta-Lucas polynomials that will be used to implement the spectral collocation method. We used a more accurate formula for the fractional derivative (Caputo sense) which is derived in a previous work. The resulting system of ODEs is transformed using the suggested method into a non-linear system of algebraic equations. The system is built as a constrained optimization problem, then optimized to obtain the series solution's unidentified coefficients. The results show that the skin-friction coefficient increases with increasing magnetic number, whereas the Casson and the local Williamson parameters exhibit reverse behavior. In addition, the effectiveness and accuracy of the proposed method are satisfied by computing the residual error function. The estimated solutions produced by using the given method were physically acceptable and accurate. • The fractional model of non-Newtonian Casson and Williamson boundary layer flow in the fluid flow are considered. • Performance was done using the strategy of the spectral collocation method of Vieta-Lucas. • Some theorems are provided in order to investigate the method's convergence. [ABSTRACT FROM AUTHOR]

• The time-fractional nonlinear diffusion wave equation with reaction and damping terms is investigated. • Performance was done using the strategy of the uniform cubic B-spline functions. • The Caputo time-fractional derivative has been estimated using the standard finite difference technique. • The convergence of the suggested blueprint is discussed in detail. The applications of the diffusion wave model of a time-fractional kind with damping and reaction terms can occur within classical physics. This quantification of the activity can measure the diagnosis of mechanical waves and light waves. The goal of this work is to predict and construct numerical solutions for such a diffusion model based on the uniform cubic B-spline functions. The Caputo time-fractional derivative has been estimated using the standard finite difference technique, whilst, the uniform cubic B-spline functions have been employed to achieve spatial discretization. The convergence of the suggested blueprint is discussed in detail. To assert the efficiency and authenticity of the study, we compute the approximate solutions for a couple of applications of the diffusion model in electromagnetics and fluid dynamics. To show the mathematical simulation, several tables and graphs are shown, and it was found that the graphical representations and their physical explanations describe the behavior of the solutions lucidly. The key benefit of the resultant scheme is that the algorithm is straightforward and makes it simple to implement as utilized in the highlight and conclusion part. [ABSTRACT FROM AUTHOR]

An inventive strategy is bestowed here to acquire the numeral roots of nonlinear boundary value problems(BVPs) of 14th-order utilizing cubic splines. Two cubic splines; Polynomial and non-polynomial, are exploited to find out the solutions of nonlinear boundary value problems(BVPs) of 14th-order. The strategies embraced in this work depend on cubic polynomial spline(CPS) and cubic non-polynomial spline(CNPS) strategy in combination of the decomposition procedure. The prescribed method transforms the boundary value problem to a system of linear equations. The algorithms we are going to develop in this paper are not only simply the approximation solution of the 14th order boundary value problems using cubic polynomial spline(CPS) and cubic non-polynomial spline(CNPS) but also describe the estimated derivatives of 1st order to 14th order of the analytic solution at the same time. These strategies will be operated on three problems to evidence the handiness of the technique by means of step size h = 1/5. The exactness of this method for detailed investigation is equated with the precise solution and conveyed through tables. To reveal the efficiency of our outcomes, the AEs (absolute errors) of the CPS and CNPS have been contrasted with Adomian Decomposition Method and Differential Transform Method and our results discovered to be more precise. • Numeral roots of nonlinear boundary value problems (BVPs) of 14th-order are investigated cubic splines. • Performance was done using the strategy of the cubic splines. • Absolute errors for cubic polynomial and non-polynomial spline have been established and compared with other numerical methods. [ABSTRACT FROM AUTHOR]

• Different wave structures for abundant solutions to the TF-BF and the STF-RLW equations are investigated. • Performance was done using the strategy of the extended tanh-function method. • Physical explanations are discussed for the obtained solutions. The general time fractional Burger- Fisher (TF-BF) and the space–time regularized long-wave (STF-RLW) equations are considered as examples of gravitational water waves in cold plasma as well as so many areas. The above equations are used in nonlinear science and engineering to study long waves in seas and harbors that travel in just one direction. First, the two equations are transformed to ODEs by applying a fractional complex transform along with characteristics of confirmable fractional derivative (CFD). Then, the extended tanh-function (ET-F) approach is investigated to find a variety of analytical solutions with different geometrical wave structures the mentioned models. The results are in the form of kink, one-, two-, multiple-solitons solutions, and other types sketched in 2D, 3D, and contour patterns. [ABSTRACT FROM AUTHOR]

In this study, the Fokas system which represents the spread of irregular pulse in monomode optical fibers is investigated via the Jacobi elliptic function expansion (JEFE) method. This method is the most powerful technique to explore solutions for a wide range of various models. As a result, different solutions such as dark–bright, singular, bright, Jacobi elliptic function, and exponential solutions are obtained. In addition, 3D and 2D graphics for the obtained solutions are investigated with the assist of a computer program by assigning particular values to the parameters involved. • New structures of optical solitons for the Fokas system were constructed. • The solutions of this model are found using the Jacobi elliptic function expansion method. • The Physical meaning for the obtained solutions is graphically investigated. [ABSTRACT FROM AUTHOR]

In this work, we study the dynamical behavior for a real physical application due to the inhomogeneities of media via analytical and numerical approaches. This phenomenon is described by the 3D Date–Jimbo–Kashiwara–Miwa (3D-DJKM) equation. For analytical techniques, three different methods are performed to get hyperbolic, trigonometric and rational functions solutions. After that, the obtained solutions are graphically depicted through 2D- and 3D-plots and numerically compared via the finite difference algorithm to check the precision of the proposed methods. • Different wave structures for abundant solutions to the (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation are investigated. • Performance was done by using the strategy of three different analytical methods. • Numerical solutions were introduced via the finite difference method. • Physical explanations are discussed for the obtained solutions. [ABSTRACT FROM AUTHOR]

This paper employs the Lie symmetry analysis to investigate novel closed-form solutions to a (2+1)-dimensional Bogoyavlenskii's breaking soliton equation. This Lie symmetry technique, used in combination with Maple's symbolic computation system, demonstrates that the Lie infinitesimals are dependent on five arbitrary parameters and two independent arbitrary functions f 1 (t) and f 2 (t). The invariance criteria of Lie group analysis are used to construct all infinitesimal vectors, commutative relations of their examined vectors, a one-dimensional optimal system and then several symmetry reductions. Subsequently, Bogoyavlenskii's breaking soliton (BBS) equation is reduced into several nonlinear ODEs by employing desirable Lie symmetry reductions through optimal system. Explicit exact solutions in terms of arbitrary independent functions and other constants are obtained as a result of solving the nonlinear ODEs. These established results are entirely new and dissimilar from the previous findings in the literature. The physical behaviors of the gained solutions illustrate the dynamical wave structures of multiple solitons, curved-shaped wave–wave interaction profiles, oscillating periodic solitary waves, doubly-solitons, kink-type waves, W-shaped solitons, and novel solitary waves solutions through 3D plots by selecting the suitable values for arbitrary functional parameters and free parameters based on numerical simulation. Eventually, the derived results verify the efficiency, trustworthiness, and credibility of the considered method. • The Lie symmetry approach is used to investigate a (2+1)-dimensional Bogoyavlenskii's breaking soliton equation. • At the outset, we derived Lie infinitesimal generators, commutator, and adjoint tables for the BBS equation. • Some exact closed-form solutions are obtained using the Lie symmetry technique. • 3D graphics are used to demonstrate the various evolutionary dynamics of some invariant solutions. • In the field of innovative research and improvement, such investigations are strongly encouraged. [ABSTRACT FROM AUTHOR]

• M−fractional derivative is used for the modeled the reaction Duffing model and diffusion–reaction equation. • Exact traveling solutions of the M−fractional generalized reaction Duffing model and density dependent M−fractional diffusion reaction equation by using three fecund, G ′ / G , 1 / G , modified G ′ / G 2 and 1 / G ′ -expansion methods. • The obtained solutions verify the all obtained solutions. Also some of the obtained solutions are explained through numerical simulations. • The obtained solutions are new and an excellent contribution in the existing scientific literature. In this paper, we obtain the exact traveling solutions of the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation by using three fertile, G ′ / G , 1 / G , modified G ′ / G 2 and 1 / G ′ -expansion methods. These methods contribute a variety of exact traveling wave solutions to the scientific literature. The obtained solutions are also verified for the aforesaid equations through symbolic soft computations. Furthermore, some results are explained through numerical simulations that show the novelty of our work. Moreover, we observe that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory. [ABSTRACT FROM AUTHOR]

• The Landau-Ginsburg-Higgs and Riemann wave equations are used to reveal phenomena.. • The impact of linearity-nonlinearity and parameters on wave profiles are evaluated. • Bright, dark, compacton, peakon, periodic, and other solitons have been extracted. • 3D and contour plots are depicted to illustrate profiles of the phenomena. The nonlinear Riemann wave equations (RWEs) and the Landau-Ginsburg-Higgs (LGH) equation are related to plasma electrostatic waves, ion-cyclotron wave electrostatic potential, superconductivity, and drift coherent ion-cyclotron waves in centrifugally inhomogeneous plasma. In this article, the interactions between the maximum order linear and nonlinear factors are balanced to compute realistic soliton solutions to the formerly stated equations in terms of hyperbolic functions. The linear and nonlinear effects rheostat the structure of the wave profiles, which vary in response to changes in the subjective parameters combined with the solutions. The established solutions to the aforementioned models using the extended tanh scheme are descriptive, typical, and consistent, and include standard soliton shapes such as bright soliton, dark soliton, compacton, peakon, periodic, and others that can be used to analyze in ion-acoustic and magneto-sound waves in plasma, homogeneous, and stationary media, particularly in the propagation of tidal and tsunami waves. [ABSTRACT FROM AUTHOR]

New optical soliton solutions for the coupled conformable Fokas–Lenells equation with spatio-temporal dispersion are obtained via Atangana's derivative operator using an integration method. The used method in this article is sine–Gordon expansion method. The obtained solutions include dark, bright, singular as well as periodic waves solitons which are graphically discussed through 2D- and 3D-plots. • The coupled conformable Fokas–Lenells equation with spatio-temporal dispersion is investigated. • The Sine–Gordon expansion method is applied to that equation. • A graphical representation of the obtained soliton solutions is presented. [ABSTRACT FROM AUTHOR]

• Different solutions for wave equations are obtained via separation of variables method and the Laplace transform. • Classical, damped and damped with source term fractional wave equations were solved. • Graphical representations are obtained for particular cases shown temporal fractality at different scales. Analytical solutions of the fractional wave equation via Caputo-Fabrizio fractional derivative are presented in this paper. For this analysis, three cases are considered, the classical, the damped and the damped with a source term defined by fractional wave equations. We show that these solutions are special cases of the time fractional equations with exponential law. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

• The Konopelchenko-Dubrovsky and the Landau-Ginzburg-Higgs equation studied using the generalized Kudryashov technique. • The obtained solutions are analyzed graphically. • Concluded that the generalized Kudryashov technique is effective for solving nonlinear evolution equations. The (2 + 1)-dimensional Konopelchenko-Dubrovsky (KD) equation and the Landau-Ginzburg-Higgs (LGH) equation describe the nonlinear waves with weak scattering and long-range interactions between the tropical, mid-latitude troposphere, the interaction of equatorial and mid-latitude Rossby waves etc. This article studies the KD and LGH models stated earlier using the generalized Kudryashov technique. We obtained a variety of analytical solutions including unknown parameters. The figures of some of the obtained solutions are sketched with certain parameters. The derived results demonstrate the efficiency and reliability of the generalized Kudryashov technique for establishing systematic solutions to nonlinear evolution equations (NLEEs). [ABSTRACT FROM AUTHOR]

In this paper, we concern ourselves with the nonlinear Kadomtsev–Petviashvili equation (KP) with a competing dispersion effect. First we examine the integrability of governing equation via using the Painlevé analysis. We next reduce the KP equation to a one-dimensional with the help of Lie symmetry analysis (LSA). The KP equation reduces to an ODE by employing the Lie symmetry analysis. We formally derive bright, dark and singular soliton solutions of the model. Moreover, we investigate the stability of the corresponding dynamical system via using phase plane theory. Graphical representation of the obtained solitons and phase portrait are illustrated by using Maple software. • The Kadomtsev–Petviashvili equation (KP) with a competing dispersion effect is investigated. • The Painlevé and Lie symmetry analysis are applied to that equation. • The stability of the corresponding dynamical system is discussed via phase plane theory. • A graphical representation of the obtained soliton solutions is presented. [ABSTRACT FROM AUTHOR]

• Different wave structures for abundant solutions to the (2 + 1)-dimensional soliton equation are investigated. • Performance was done using the strategy of the double (G'/G, 1/G)-expansion method. • Physical explanations are discussed for the obtained solutions. In this paper, the double ( G ' / G , 1/ G)-expansion method is employed to establish explicit general solutions of the nonlinear (2 + 1)-dimensional soliton equation. A variety of exact travelling wave solutions are attained involving three functions that are classified into rational, trigonometric and hyperbolic with especial parameters that originate the solitary explicit and new periodic solutions. The used method is the generalization of the ( G ' / G)-expansion method and it rediscovers all the acquainted solitary wave solutions that are obtained by means of the ( G ' / G)-expansion method. [ABSTRACT FROM AUTHOR]

In this work, we research the mixed lump–kink solutions and their dynamic properties of the extended (3+1)-dimensional Jimbo–Miwa equation. Mixed lump–kink solutions are triggered by the interaction between lump soliton and exponential function, the interaction between lump soliton and hyperbolic function, and the interaction between lump soliton and periodic wave. Their dynamics properties are shown by means of many graphics and corresponding density plots. • The extended (3+1)-dimensional Jimbo–Miwa equation is researched. • The interaction between lump soliton and exponential function is obtained. • The interaction between lump soliton and hyperbolic function is presented. • The interaction between lump soliton and periodic wave is investigated. • Their dynamics properties are shown by means of many graphics and corresponding density plots. [ABSTRACT FROM AUTHOR]

• Stable solitary solutions have been developed for NLEEs related to optics and superconductivity. • The effects of linearity and nonlinearity on the dynamics of wave profiles have been interpreted. • Standard wave profiles including kink, bell-shape soliton, and periodic solitons have been found. • The 3D and contour plots are depicted to illustrate profile of the phenomena. In the field of quantum mechanics and fluid physics, especially in the study of nonlinear geometric optics and superconductivity, the Landau-Ginzburg-Higgs (LGH) and the (2 + 1)-dimensional Novikov-Veselov (NV) equations are two significant models. In this article, we have affirmed that wave profile changes with the change of the free parameters associated with them and are mainly dominated by linear effects. The effects of nonlinearity and wave speed on the wave contours have also been analyzed. On account of this, wave solutions are computed concerning rational, hyperbolic, and trigonometric structures balancing the exponents of linear and nonlinear terms of the highest order, from which scores of typical wave profiles including kink, bell-shape soliton, lump, and periodic waves have been extracted. The wave solutions are designated through extending the typical concept of the sine-Gordon expansion (SGE) method from the lower dimensional to the higher dimensional nonlinear evolution equations. This study establishes the capability of the stated method in solving both lower and higher dimensional nonlinear evolution equations. The solutions are analyzed by sketching figures for different values of related variables, and it is observed that the attributes of these solutions are pivotal in the selection of parameters. [ABSTRACT FROM AUTHOR]

In this work, the analytic solutions for different types of nonlinear partial differential equations are obtained using the multiple Exp-function method. We consider the stated method for the (3+1)-dimensional generalized shallow water-like (SWL) equation, the (3+1)-dimensional Boiti–Leon- Manna–Pempinelli (BLMP) equation, (3+1)-dimensional generalized variable-coefficient B-type Kadomtsev–Petviashvili (VC B-type KP) equation and the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation. We obtain multi classes of solutions containing one-soliton, two-soliton, and triple-soliton solutions. All the computations have been performed using the software package Maple. The obtained solutions include three classes of soliton wave solutions in terms of one-wave, two-waves, and three-waves solutions. Then the multiple soliton solutions are presented with more arbitrary autocephalous parameters, in which the one, two, and triple solutions localized in all directions in space. Moreover, the obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. The different types of obtained solutions of aforementioned nonlinear equations arising in fluid dynamics and nonlinear phenomena. • Different solutions for NPDEs are obtained via the multiple Exp-function method. • Many solutions containing one-, two-, and triple-soliton solutions are investigated. • All the computations have been performed using the software package Maple. • The obtained solutions are presented with more arbitrary autocephalous. [ABSTRACT FROM AUTHOR]

• Abundant solutions of the electrical transmission lines in mZK equation are found. • Performance was done using the strategy of two different techniques. • Physical explanations are discussed for the obtained solutions. Discrete networks, nonlinear networks, high speed computer data buses, computer network connections, and cable signal distribution are analyzed using the modified Zakharov-Kuznetsov (mZK) equation as a model governing the propagation in the electrical transmission lines. In this study, we establish viable extensive traveling wave solutions with the familiar singular kink type, kink type, bell-type solution, singular bell-type solution, t-type solution, and some advanced soliton solutions through the modified simple equation (MSE) and the sine-Gordon expansion methods. The established solutions are highly stable, resilient and capable to travel long distance. The solutions obtained can determine the voltage and current flow that can be used to design the transmission lines. The density plot and three-dimensional diagrams of the solutions obtained illustrate the voltage and current distributions in the transmission line. The traveling wave solutions are found in closed-form which simplify their utilization. The implemented methods are compatible for the extraction of traveling wave solutions. [ABSTRACT FROM AUTHOR]

• The broad-ranging general stable soliton solutions to some nonlinear equations have been established associated with parameters and integral constants in physics and gas dynamics. • The characteristics of gas flow ensuing shock fronts and nonlinear optics have been analysed. • The 3D and contour plots of the obtained solutions are depicted to illustrate profile of the phenomena. Stable soliton solutions for the nonlinear Klein–Gordon equation in condensed matter physics, particle physics, nonlinear optics, solid state physics and the gas dynamics equation ensuing in shock fronts have been established by putting use of the sine-Gordon expansion procedure. We establish the broad-ranging familiar stable wave solutions related with some free parameters, for particular values of which some of the subsisting solutions in the literature are re-established and several fresh solutions are formulated. The contour and 3D shapes are depicted to describe the physical phenomena of the established solutions. Scores of solitary wave solutions are obtained such as, compacton, smooth soliton, anti-bell type solitary wave, bell-type solitary wave, kink and some other new types of solitary wave solutions. The obtained results might play important role in optics, quantum mechanics, mathematical physics and engineering. It is noticeable to inspect that the sine-Gordon expansion procedure is an efficient, competent and straightforward mathematical tool to ascertain exact solitary wave solutions. [ABSTRACT FROM AUTHOR]

• Different wave structures for abundant solutions to the two-mode Sawada-Kotera equation are investigated. • Performance was done using the strategy of the modified Kudryashov and new auxiliary equation methods. • Physical explanations are discussed for the obtained solutions. New dual-wave soliton solutions are addressed for the two-mode Sawada-Kotera (TmSK) equation arising in fluids by the modified Kudryashov and new auxiliary equation methods. As outcomes, bright, dark, periodic, and singular-periodic dual-wave solutions are obtained. The graphs of the solutions are provided to show the impact of the parameters. A comparison between our solutions and the existing ones is carried out. [ABSTRACT FROM AUTHOR]

• Different wave structures for abundant solutions to the DJKM model are investigated. • Performance was done using the strategy of the Hirota's bilinear method. • Physical explanations are discussed for the obtained solutions. In this work, N-soliton waves, fusion solutions, mutiple M-lump solutions and the collision phenomena between one-M-lump and one-, two-soliton solutions to the (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation are successfully revealed. A class of one-, two-, three-soliton, one-, two-fusion solutions are derived via the Hirota bilinear method and 1-M-lump, 2-M-lump solutions are constructed via the long-wave method. Moreover, physical collision phenomenon of 1-M-lump with one-, two-soliton solutions and also, with fusion solutions are successfully presented. The velocity of the 1-M-lump wave in x- and y-direction are also studied. [ABSTRACT FROM AUTHOR]