Extended Fan Sub-equation Method Research Articles

On some soliton structures to the Schamel–Korteweg-de Vries model via two analytical approaches

The Schamel–Korteweg-de Vries (S-KdV) model is used to predict the influence of surface for deep water in the presence of solitary waves. The aim of the study is to study the governing model analytically by employing the extended modified auxiliary equation mapping approach and the extended FAN sub-equation method. The 3D, 2D and contour plots are drawn to demonstrate the physical nature of the nonlinear model for a set of parameters. As a result, dark solitons, light solitons, solitary waves, periodic solitary waves, rational functions, and elliptic function solutions are established. Furthermore, the the developed results are verified with the aid of latest computing tool such as Mathematica or Maple. The applied strategy appears to be a more powerful and efficient scheme for achieving exact solutions to a number of diversified contemporary models of recent eras.

Construction of optical pulses and other solutions to optical fibers in absence of self-phase modulation

This paper investigates the optical solitons to the Biswas–Arshed equation (BAE) with third-order dispersion and self-steepening coefficients that communicate pulse propagation in fiber optics. The equation is considered where self-phase modulation is negligibly small and hence removed. Under different constraint conditions, the extended Fan sub-equation method (EFSEM) is used to derive the different kinds of solutions in the forms of bright, dark, singular, complex and combined solitons. In addition, Jacobi elliptic function (JEF) periodic, hyperbolic-type solutions are secured. This technique is useful for solving NLPDEs because it combines the results of numerous operations to provide previously extracted solutions as well as fresh solutions. While describing the physical representation of particular solutions, we also plot 3D, and 2D graphs by picking appropriate values for the solutions’ related parameters. On the basis of a comparison of our results with well-known ones, the study indicates that our answers are innovative. Nonlinear dynamical behavior of a particular system can be improved and the methodology applied can be shown to be effective by these findings. Engineering modelers will greatly benefit from this study, according to our opinion. Based on the results obtained, the computational method adopted is efficient and direct, as well as succinct and can be applied to complex systems.

Propagation of diverse exact solitary wave solutions in separation phase of iron (Fe-Cr − X(X = Mo, Cu)) for the ternary alloys

In this paper, a specific consideration is paid to the nonlinear dynamics of solitary waves to kinetics of phase separation in iron [Formula: see text] based on ternary alloys. The convective–diffusive Cahn–Hilliard (CH) equation is used as a mathematical model to describe the dynamics of the separation phase for the ternary alloys of iron. A variety of solitary wave solutions with unknown parameters are extracted in different shapes like kink-type, bell-shape, shock-type, combine soliton, trigonometric, hyperbolic and Jacobi’s elliptic function solutions with the assistant of recently computational tools, namely, extended Fan-sub equation method (EFSEM) and extended auxiliary equation method (EAEM). In addition, 3D, 2D, and their corresponding contour profiles of earned results are sketched in order to observe their dynamics with the choices of involved parameters. On the bases of achieved results, we may claim that the proposed computational methods are direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations.

Application of the Extended Fan Sub-Equation Method to Time Fractional Burgers-Fisher Equation

Abstract In this paper, the extended Fan sub-equation method to obtain the exact solutions of the generalized time fractional Burgers-Fisher equation is applied. By applying this method, we obtain different solutions that are benefit to further comprise the concepts of complex nonlinear physical phenomena. This method is simple and can be applied to several nonlinear equations. Fractional derivatives are taken in the sense of Jumarie’s modified Riemann-Liouville derivative. A comparative study with the other methods approves the validity and effectiveness of the technique, and on the other hand, for suitable parameter values, we plot 2D and 3D graphics of the exact solutions by using the extended Fan sub-equation method. In this work, we use Mathematica for computations and programming.

Construction of new exact solutions of the resonant fractional NLS equation with the extended Fan sub-equation method

The fucus of this study is to find a set of some novel solutions concerning the resonant fractional nonlinear Schrödinger equation (R-FNLSE) with quadratic-cubic nonlinearity by employing the extended FAN sub-equation approach. The parameter α∈(0,1] is the core constraint which simulate the flow rate propagation, plays a key rote in telecommunications and the theory of optical fibres. This equation expresses the gesture of solitons and Madelung fluids in various nonlinear systems. These outcomes are optical, bright, dark, explicit, periodic and combined wave solutions and efficiently demonstrated with the aid of 3D plots. The representation of these solutions is carried out in order to have an idea about the structure of such model and can also be extended to many other complex models of recent era.

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

Traveling, periodic and localized solitary waves solutions of the (4+1)-dimensional nonlinear Fokas equation

By utilizing the three distinctive approaches specifically, the extended Fan sub-equation method, the exp[ $$-G(\xi )$$ ]-function expansion method, and the fractional transformation method, the traveling waves and soliton solutions of the (4+1)-dimensional nonlinear Fokas equation are extracted. Meanwhile, some parametric constraint conditions are described. The acquired solutions are singular and nonsingular soliton solutions, periodic solutions, breather solution, rational solutions, trigonometric periodic wave solutions, hyperbolic solutions, Weierstrass, and Jacobi elliptic doubly periodic wave solutions. The dynamics of some of the obtained solutions are investigated and described in 2-dimensional figures by choosing appropriate parameter values. The comparison of our obtained results with the other solutions in literature shows that the obtained solutions of this paper are new and have not been formulated before by other techniques. We believe that all these results are useful to enrich the knowledge of the important physical phenomenon characterized by the Fokas equation. The reported solutions illustrate the straightforwardness, reliability, and effectiveness of the used techniques that can be further employed to higher-dimensional nonlinear evolution equations.

Modulation instability analysis, optical and other solutions to the modified nonlinear Schrödinger equation

This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.

Investigation of optical solitons in birefringent polarization preserving fibers with four-wave mixing effect

The paper studies the optical solitons with coupled nonlinear Schrödinger system (CNLSS) that describes the propagation of waves in birefringence polarization-preserving fibers with four-wave mixing effect. The system is discussed with the effects of group velocity dispersion, self-phase and cross-phase modulations. The mechanism that is used to investigate is the extended Fan-sub equation method. It is observed that this system exhibits a collection of rich solitons (single and combined) for different values of parameters. The 3D figures are also plotted to show the physical behavior of obtained solutions under different constraint conditions.

Investigation of soliton solutions with different wave structures to the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation

The principal objective of this article is to construct new and further exact soliton solutions of the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets and explains their ordering in ferromagnetic materials. These solutions are exerted via the new extended FAN sub-equation method. We successfully obtain dark, bright, combined bright-dark, combined dark-singular, periodic, periodic singular, and elliptic wave solutions to this equation which are interesting classes of nonlinear excitation presenting spin dynamics in classical and semi-classical continuum Heisenberg systems. 3D figures are illustrated under an appropriate selection of parameters. The applied technique is suitable to be used in gaining the exact solutions of most nonlinear partial/fractional differential equations which appear in complex phenomena.

A variety of exact solutions to (2+1)-dimensional schrödinger equation

ABSTRACT The article investigates that the -dimensional Schrödinger equation admits a rich variety of exact solutions with the range of five parameters. The solutions are of qualitatively different nature, depending on the different values of parameters. The obtained solutions are of triangular-type, soliton-type, doubly periodic-like, single and combined non-degenerate jacobi elliptic function like. Additionally, the constraint conditions for the existence of the solutions are also emerged during the derivation. The important fact of using the extended Fan sub-equation method is to take the full advantage of clear relationship among general elliptic equation involved with five parameters and other existing sub-equations involved with three parameters.

Optical solitons with quadratic–cubic nonlinearity and fractional temporal evolution

The paper studies the single and combined optical solitons in the physical model of telecommunication system that describes the dynamics of optical solitons with quadratic–cubic nonlinearity in the mitigate internet bottleneck. This fractional temporal evolution is considerd with the coefficients of self-steeping, group velocity, inter-model and nonlinear dispersions. The extended Fan sub-equation method is used to investigate the families of single optical solitons (bright, dark and singular) for the different values of parameters.

Explicit, periodic and dispersive optical soliton solutions to the generalized nonlinear Schrödinger dynamical equation with higher order dispersion and cubic-quintic nonlinear terms

In this article, the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms have been discussed analytically using extended Fan sub-equation method. The results hold numerous traveling wave solutions like optical, bright, dark, explicit, periodic and combined wave solutions with the aid of five parameters that are of key importance in elucidating some physical circumstance.

Dispersive traveling wave solutions to the space–time fractional equal-width dynamical equation and its applications

In this article, some new traveling wave solutions to the space–time fractional equal-width equation are constructed with the help of the extended Fan sub-equation method. A simple transformation is introduced to convert the fractional order partial differential equation into an ordinary differential equation. As a result, the bright, dark, singular and combined wave solitons are observed for different values of two parameters. Moreover, the graphical representations are also depicted.

Application of extended Fan sub-equation method to $$\mathbf {(1+1)}$$ ( 1 + 1 ) -dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation with fractional evolution

The article studies the application of the extended Fan sub-equation method to \((1+1)\)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation with fractional evolution. This equation describes the hydromagnetic waves in cold plasma, acoustic waves in inharmonic crystals and acoustic gravity waves in compressible fluids. The structure of the extended Fan sub-equation method on the basis of time-fractional derivative is presented. The main idea of the method is to take full advantage of the general elliptic equation involving five parameters. The fractional derivatives are taken as in the sense of Jumarie’s modified Riemann–Liouville derivative. The method is reliable and gives more general exact solutions than the existing methods.

Bright, dark and other optical solitons with second order spatiotemporal dispersion

In this article, the single (bright, dark, singular) and combined (bright-dark, dark-singular) optical solitons have been extracted using extended fan sub-equation method. The corresponding integrability criteria, also known as constraint conditions, naturally emerge during the analysis of this model which is considered with the coefficients of group velocity dispersion and second order spatiotemporal dispersion.

Dark and singular optical solitons perturbation with fractional temporal evolution

The article studies the dynamics of dark, singular, combined optical solitons and many other periodic solutions to fractional temporal perturbed nonlinear Schrödinger equation in nonlinear optics. The fractional extended Fan sub-equation method is first time used for any fractional temporal nonlinear Schrödinger equation. The solutions are of qualitatively different nature, depending on the five parameters. The constraint conditions, for the existence of the solitons, are also listed. Moreover a couple of other solutions known as combined soliton and combined periodic solution, fall out as a by product in limiting cases.

New and more general traveling wave solutions for nonlinear Schrödinger equation

An extended Fan sub-equation method is used to seek some new and more general traveling wave solutions of nonlinear Schrödinger equation (NLSE). The important fact of this method is to take the full advantage of clear relationship among general elliptic equation involving five parameters and other existing sub-equations involving three parameters. It is preferable to use this method to solve NLSE because this method gives us all the solutions obtained previously by the application of at least four methods (the method of using Riccati equation, or auxiliary ordinary differential equation method, or first kind elliptic equation or the generalized Riccati equation as mapping equation) in a unified manner. So it is shown that this method is concise and its applications are promising.

Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the Davey-Stewartson Equations

We used what we called extended Fan's sub-equation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the Davey-Stewartson (DS) equations. The basic idea of the proposed extended Fan's subequation method is to take fulls advantage of the general elliptic equations, involving five parameters, which have many new solutions and whose degeneracies lead to special subequations involving three parameters like Riccati equation, first-kind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the Davey-Stewartson (DS) equations including more general soliton solutions, triangular solutions, and double-periodic solutions are constructed by symbolic computation.

A further improved extended Fan sub-equation method for (2+1)-dimensional breaking soliton equations

In this paper, a further improved extended Fan sub-equation method is used to construct exact solutions of the (2+1)-dimensional breaking soliton equations. As a result, many new and more general non-travelling wave and coefficient function solutions are obtained including soliton-like solutions, triangular-like solutions, single and combined non-degenerate Jacobi elliptic wave function-like solutions, Weierstrass elliptic doubly-like periodic solutions, each of which contains an arbitrary function of two variables. The results show the proposed method give new and more general exact solutions. More importantly, the method with the aid of symbolic computation provides a very effective and powerful mathematical tool for solving a great many non-linear partial differential equations in mathematical physics.