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

This research paper investigates the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony equation. The new Kudryashov and generalized Arnous methods are employed to obtain the generalized solitary wave solution. The phase plane theory examines the bifurcation analysis and illustrates phase portraits. Finally, the external perturbation terms are considered to reveal its chaotic behavior. These findings contribute to a deeper understanding of the dynamics of the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony wave equation and its applications in real-world phenomena. [ABSTRACT FROM AUTHOR]

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory solitary wave propagation in dispersive media. The similarity reductions and explicit solutions are derived by using Lie symmetry method. Further, using Ibragimov's theorem conservation laws for this equation are discussed. [ABSTRACT FROM AUTHOR]

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

This paper analyzes the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation, which appears in the fields of applied magnetism and nanophysics. By taking advantage of two mathematical integration approaches, namely, the modified generalized exponential rational function method and the extended tanh function method, a variety of exact optical soliton solutions are obtained for the governing cmKdV equation. These acquired soliton solutions are determined in terms of hyperbolic, exponential, and trigonometric function types. By choosing suitable values of parameters, some 3D, 2D, and contour plots are portrayed with the aid of symbolic computation in Mathematica to visualize the underlying dynamics of the generated solutions. These solutions include doubly soliton, multi-soliton, singular periodic soliton, anti-bell-shaped soliton, and hyperbolic structures. Moreover, the modulation instability of the governing equation is also investigated by using the linear stability analysis. The results presented in this paper are novel and are reported for the first time in the literature. Again, modulation instability analysis was carried out on the governing model for the first time. Thus, the results obtained demonstrate that the two new mathematical schemes are quite concise and effective and can be useful in understanding the dynamical behaviors of many other nonlinear physical models appearing in nonlinear optics, nanophysics, and so many other areas of nonlinear sciences. [ABSTRACT FROM AUTHOR]

In this work, we use the generalized Riccati equation mapping method (GREMM) to seek solitary and periodic wave solutions to the generalized complex coupled Schrödinger–Boussinesq equations, which is a more useful model for describing the stationary propagation of coupled nonlinear upper-hybrid and magnetosonic waves in magnetized plasma. The GREMM approach is applied to compute a variety of soliton and other solutions in distinctive forms such as trigonometric, hyperbolic, singular, and interaction solutions in addition to rational. The main feature of this work is the achieved soliton solutions, which were not before constructed and are more general. All of the obtained solutions are validated by re-inserting them into the governing system with the help of mathematica. Moreover, 3D and 2D combined line profiles, as well as their contour graphics, are plotted to depict the dynamical behaviors of the retrieved solutions using numerical simulations. We exhibit how the wave profiles are impacted with varying time. Computational results demonstrate that applied research methodology and analytical results enrich the dynamics of the governing model. [ABSTRACT FROM AUTHOR]

In this article, we study a model problem for the advection–reaction–diffusion equation involving a new nonsingular time‐fractional derivative with Rabotnov fractional‐exponential (RFE) kernel. In order to solve this model numerically, we first obtain the numerical approximation of RFE fractional derivative for a simple polynomial function, which gives rise to an operational matrix of fractional differentiation. We illustrate the accuracy and validity of this operational matrix with the aid of an example. We use Legendre collocation technique together with the newly developed operational matrix to find the numerical solution of the given model. The numerical results depict the feasibility and efficacy of our method. The error estimates show that our method is valid with great accuracy and is applicable to a fractional ODE system and an integral equation with RFE kernel fractional derivative. [ABSTRACT FROM AUTHOR]

The prime objective of this paper is to obtain some new families of exact solitary wave solutions of the Klein–Gordon–Zakharov (KGZ) equations via computerised symbolic computation on Wolfram Mathematica. By applying the generalised exponential rational function method, numerous exact soliton solutions are constructed for the KGZ equations, which provide a model of the interaction between the Langmuir wave and the ion-acoustic wave in high-frequency plasma. Consequently, the exact solitary wave solutions are obtained in different forms of dynamical wave structures of solitons including multisolitons, lump-type solitons, travelling waves, kink waves, also trigonometric and hyperbolic function solutions, and rational function solutions. Moreover, the dynamical behaviour of the resulting multiple soliton solutions is discussed both analytically and graphically by using suitable values of free parameters through numerical simulation. The reported results have rich physical structures that are helpful to explain the nonlinear wave phenomena in plasma physics and soliton theory. [ABSTRACT FROM AUTHOR]

The Lie symmetry method is used to obtain a variety of closed-form wave solutions for the extended (3 + 1)-dimensional Jimbo–Miwa (JM) Equation, which describes certain interesting higher-dimensional waves in ocean studies, marine engineering, and other fields. By applying the Lie symmetry technique, we explicitly investigate all the possible vector fields, commutation relations of the considered vectors, and various symmetry reductions of the equation. Based on three stages of Lie symmetry reductions, the JM equation is reduced to several nonlinear ordinary differential equations (NLODEs). Consequently, abundant closed-form wave solutions are achieved, including arbitrary functional parameters. Evolutionary dynamics of some analytic wave solutions are demonstrated through three-dimensional plots based on numerical simulation. Consequently, singular soliton, kink waves, periodic oscillating wave profiles, combined singular soliton profiles, curved-shaped multiple solitons, and periodic multiple solitons with parabolic wave profiles are demonstrated by taking advantage of symbolic computation work. The obtained analytical wave solutions, which include arbitrary independent functions and other constants of the governing equation, could be used to enrich the advanced dynamical behaviors of solitary wave solutions. Furthermore, the study of conservation laws is investigated via the Ibragimov technique for Lie point symmetries. [ABSTRACT FROM AUTHOR]

In this work, the new optical soliton solutions and interaction solutions for the space-time fractional Fokas–Lenells equation with fractional M -derivatives are constructed via three mathematical analytical techniques, namely the extended SE method, unified solver method, and three-wave methods. The results have proved the efficiency of the suggested techniques for obtaining abundant optical soliton solutions to nonlinear evolution equations (NLEEs) and closed-form solutions in the forms of rational function solutions; hyperbolic and trigonometric function solutions and multi-wave interaction solutions are obtained. These techniques are more efficient, robust, and powerful mathematical tools for acquiring several optical soliton solutions for many other fractional space-time NLEEs that arise in optical physics and plasma physics. The graphical representations of the combined optical solitons are demonstrated using three- and two-dimensional graphics. [ABSTRACT FROM AUTHOR]

• Mittag-Leffler kernel derivative. • Advection-reaction-diffusion equation with fractional order. • fractional differentiation. • Legendre operational matrix computations. This paper presents the fractional formulation and numerical solution of a non-linear fractional diffusion equation with advection and reaction terms. The fractional derivative employed in this equation is of non-singular type with Mittag-Leffler kernel. To investigate this fractional equation numerically, we adopted a collocation method with the Legendre operational matrix. The numerical value of Atangana-Baleanu Caputo (ABC) fractional differentiation of polynomial function f (ϑ) = ϑ k is determined by using the definition of Mittag-Leffler kernel fractional differentiation. The operational matrix of fractional differentiation of Mittag-Leffler kernel fractional derivative is determined numerically employing a polynomial function. Within the above context, we solve the fractional equation in-hand by using this matrix and using the collocation method based upon Legendre polynomial. The effectiveness of these techniques is illustrated through different cases that include fractional diffusion, advection-diffusion, and reaction-diffusion equation. The feasibility of the proposed derived ABC fractional differentiation operational matrix is clearly demonstrated by validating the numerical resolutions against exact solutions. The results demonstrated that the method is capable of resolving the non-linear fractional diffusion equation with different fractional order. [ABSTRACT FROM AUTHOR]

In this work, computational analysis of generalized Burger's-Fisher and generalized Burger's-Huxley equation is carried out using the sixth-order compact finite difference method. This technique deals with the nonstandard discretization of the spatial derivatives and optimized time integration using the strong stability-preserving Runge-Kutta method. This scheme inculcates four stages and third-order accuracy in the time domain. The stability analysis is discussed using eigenvalues of the coefficient matrix. Several examples are discussed for their approximate solution, and comparisons are made to show the efficiency and accuracy of CFDM6 with the results available in the literature. It is found that the present method is easy to implement with less computational effort and is highly accurate also. [ABSTRACT FROM AUTHOR]

In this paper, we present a work on dynamic equation of Deoxyribonucleic acid (DNA) derived from the Peyrard–Bishop (PB) model oscillator chain for various dynamical solitary wave solutions. In order to construct novel soliton solutions in the DNA dynamic PB model with beta-derivative, the efficiency of the newly developed algorithms is being investigated, which could include a new auxiliary equation strategy (NAES). Some precise soliton solutions comprising dark, light and other forms of multi-wave soliton solutions are achieved via the proposed methods. Furthermore, mathematical models demonstrate the singularity of our work in comparison to current literary materials and even describe some results using the classic Peyrard–Bishop model. All the established results contribute to the possibility of extending the approach to solve other nonlinear equations of fractional space–time derivatives in nonlinear sciences. The strategy that has been proposed recently is specific and is being employed to produce novel closed-form solutions for all many other FNLEEs. [ABSTRACT FROM AUTHOR]

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods' performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order. [ABSTRACT FROM AUTHOR]

The prime objective of this paper is to obtain the exact soliton solutions by applying the two mathematical techniques, namely, Lie symmetry analysis and generalized exponential rational function (GERF) method to the (2+1)-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili (g-CHKP) equation. First, we obtain Lie infinitesimals, possible vector fields, and commutative product of vectors for the g-CHKP equation. By the means of symmetry reductions, the g-CHKP equation reduced to various nonlinear ODEs. Subsequently, we implement the GERF method to the reduced ODEs with the help of computerized symbolic computation in Mathematica. Some abundant exact soliton solutions are obtained in the shapes of different dynamical structures of multiple-solitons like one-soliton, two-soliton, three-soliton, four-soliton, bell-shaped solitons, lump-type soliton, kink-type soliton, periodic solitary wave solutions, trigonometric function, hyperbolic trigonometric function, exponential function, and rational function solutions. Consequently, the dynamical structures of attained exact analytical solutions are discussed through 3D-plots via numerical simulation. A comparison with other results is also presented. [ABSTRACT FROM AUTHOR]

In the present article, our main aim is to construct abundant exact solutions for the (2 + 1) -dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM) equation by using two powerful techniques, the Lie symmetry method and the generalised exponential rational function (GERF) method with the help of symbolic computations via Mathematica. Firstly, we have derived infinitesimals, geometric vector fields, commutation relations and optimal system. Therefore, the KP-BBM equation is reduced into several nonlinear ODEs under two stages of symmetry reductions. Furthermore, abundant solutions are obtained in different shapes of single solitons, solitary wave solutions, quasiperiodic wave solitons, elastic multisolitons, dark solitons and bright solitons, which are more relevant, meaningful and useful to describe physical phenomena due to the existence of free parameters and constants. All these generated exact soliton solutions are new and completely different from the previous findings. Moreover, the dynamical behaviour of the obtained exact closed-form solutions is analysed graphically by their 3D, 2D-wave profiles and the corresponding density plots by using the mathematical software, which will be comprehensively used to explain complex physical phenomena in the fields of nonlinear physics, plasma physics, optical physics, mathematical physics, nonlinear dynamics, etc. [ABSTRACT FROM AUTHOR]

The main motive of this article is to study the recently developed Atangana‐Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag‐Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two‐dimensional cable equation in which time‐fractional derivative is of Mittag‐Leffler kernel type. First, we derive an approximation formula of the fractional‐order ABC derivative of a function tk using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two‐dimensional model of the time‐fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two‐dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag‐Leffler fractional derivative. [ABSTRACT FROM AUTHOR]

In this research work, we study the model of nonlinear reaction–diffusion equation, diffusion‐wave equation, and Cattaneo equation with the help of a numerical method in which time‐fractional derivative is of Caputo–Fabrizio type (C‐F). First, we derive a formula of the fractional‐order C‐F derivative of xk. Using this formula and some properties of shifted Chebyshev polynomials, we find out the operational matrix of the C‐F derivative. We solve a new class of fractional partial differential equations (FPDEs) using the collocation method and C‐F operational matrix. We depict the feasibility and validity of this numerical method by giving the solution of some numerical examples. We take the diffusion equation, nonlinear reaction–diffusion equation, diffusion‐wave equation, and Cattaneo equation as a particular case of our model. By comparing the obtained numerical results with the given exact solution, we draw an error table depicting that our results are so accurate and desirable accuracy. In the authors' knowledge, the C‐F shifted Chebyshev operational matrix is derived for the first time in this work. [ABSTRACT FROM AUTHOR]

Our motive in this contribution is to find out the operational matrix of fractional derivative having non singular kernel namely Rabotnov fractional-exponential (RFE) kernel which is recently introduced and seeking numerical solution of non-linear Lienard equation which have Rabotnov fractional-exponential kernel fractional derivative. First we derive an approximation formula of the fractional order derivative of polynomial function z k in term of RFE kernel. Using this formula and some properties of shifted Legendre polynomials, we find out the operational matrix of fractional order differentiation. In the author of knowledge this operational matrix of RFE kernel fractional derivative is derived first time. We solve a new class of fractional partial differential equation (FPDEs) by implementation of this newly derived operational matrix. We show that our newly derived operational matrix is valid by taking an fractional derivative of a polynomial. Also, we study a new model of Lienard equation with RFE kernel fractional derivative and we can easily predict the feasibility of our numerical method to this new model. [ABSTRACT FROM AUTHOR]

In this work, Lie symmetry analysis and one-dimensional optimal system for Pavlov equation are presented. All the possible vector fields, their commutative and adjoint relations are carried out under invariance property of Lie group theory. On the basis of optimal system, similarity reductions of Pavlov equation are obtained. A repeated process of similarity reductions transforms the Pavlov equation into ordinary differential equations, which generate invariant solutions. The obtained invariant solutions are supplemented by numerical simulation to analyse the physical behaviour. Thus, their parabolic, multisoliton, nonlinear, kink and antikink wave profiles are traced in results and discussions sections. [ABSTRACT FROM AUTHOR]

In the present scientific work, an operational matrix scheme with Laguerre polynomials is applied to solve a space-time fractional order non-linear Cahn-Hilliard equation, which is used to calculate chemical potential and free energy for a non-homogeneous mixture. Constructing operational matrix for fractional differentiation, the collocation method is applied to convert Cahn-Hilliard equation into an algebraic system of equations, which have been solved using Newton method. The prominent features of the manuscript is to providing the stability analysis of the proposed scheme and the pictorial presentations of numerical solution of the concerned equation for different particular cases and showcasing of the effect of advection and reaction terms on the nature of solute concentration of the considered mathematical model for different particular cases. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) equation are constructed which appear in the magnetized two-ion-temperature dusty plasma and quantum physics. Lie group of transformation method is proposed to investigate the solution of (3+1)-dimensional eZK equation via Lie symmetry method. The optimal system of one dimensional Lie subalgebra is constructed by using Lie point symmetries. The three dimensional eZK equation reduced into number of ordinary differential equations (ODEs) by applying similarity reductions. Consequently, solutions so extracted are more general than erstwhile known results. We have obtained twenty one solutions in the explicit form, some of them are likewise general and some are new for the best study of us. Eventually, single soliton, quasi-periodic soliton, multisoliton, lump-type soliton, traveling wave and solitary wave-interaction behavior are illustrated graphically through numerical simulation for physical affirmation of the results. Please check whether the affiliations are correct. [ABSTRACT FROM AUTHOR]

In this work, we studied time fractional Biswas–Milovic equation (FBME) which describes the long distance optical communication with Kerr law nonlinearity. We analyzed group analysis of equation by using Lie symmetry analysis and reduced it to fractional ordinary differential equation. We obtained analytic soliton solutions for the equation by means of D t α G G expansion method in form of hyperbolic, trigonometric and rational function form. The conservation laws for fractional governing equation are constructed by using non linear self adjointness and Noether's operators. The numerical approximated solution are also obtained by using residual power series method. Some figures are also added to understand the solutions. [ABSTRACT FROM AUTHOR]