Generalized Hirota Satsuma Coupled KdV System Research Articles

New periodic solutions and solitary wave solutions for the time-fractional differential equations

In this paper, we obtain many different types of exact solutions to the time-fractional Klein–Gordon equation and the time-fractional generalized Hirota-Satsuma coupled KdV system by using the modified rational function approach. Some new solutions such as the kink-periodic solution, the anti-kink-periodic solution and the concave-convex-periodic solution are constructed. Furthermore, the kink and the singular kink waves, the bell shaped soliton and the singular soliton solutions of the two equations also are found. Some numerical simulations are presented, these works can effectively reflect the propagation phenomena of time-fractional nonlinear systems, and also enable us to understand time-fractional nonlinear physical phenomena more clearly.

New analytical technique for prototype closed form solutions of certain nonlinear partial differential equations

This work proposes a novel modified (G’/G2)-expansion method to construct the exact travelling wave solutions of three famous nonlinear partial differential equations namely, nanoionic currents along microtubules equation, Benney-Luke equation and the generalized Hirota-Satsuma coupled KdV system. In the result, these solutions can be divided into different types i.e. trigonometric, rational and exponential solutions. The computational tool Maple is used to arrive at these solutions. Applying the novel modified (G’/G2)-expansion approach yields results that are trustworthy, easy to understand, and useful in a variety of disciplines including physics, biology, finance, engineering, statistics, and more. Software is used to create the two- and three-dimensional graphs. The comparison of the obtained solutions is also given in the form of Tables 1, 2 and 3, to show the novelty of our obtained solutions.

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

In this paper, two types of fractional nonlinear equations in Caputo sense, time-fractional Newell–Whitehead equation (FNWE) and time-fractional generalized Hirota–Satsuma coupled KdV system (HS-cKdVS), are investigated by means of the q-homotopy analysis method (q-HAM). The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions. Due to the presence of the auxiliary parameter h in this method, just a few terms of the series solution are required in order to obtain better approximation. For the sake of visualization, the numerical results obtained in this paper are graphically displayed with the help of Maple.

Diverse solitary wave solutions of fractional order Hirota-Satsuma coupled KdV system using two expansion methods

The generalized Hirota-Satsuma coupled KdV system with fractional-order derivative plays a significant role to simulate the interaction of nearby identical-weight particles in a crystal lattice structure, two long waves interaction with different dispersion relationships, the description of shallow-water wave propagation, ion-acoustic waves, plasma physics and some other fields. In this study, diverse analytical wave solutions in general and standard structures are established of the stated equation by introducing two viable techniques, namely, the generalized Kudryashov scheme and the two variables G'/G,1/G-expansion approach. The solutions are established in terms of elementary functions, specifically trigonometric functions, rational functions, exponential functions, and hyperbolic functions. For definite values of free parameters, the obtained analytical wave solutions transform into solitary wave solutions. The graphical patterns of the wave solutions with there-, two-, and contour plots are depicted magnificently to elucidate the internal structure of the phenomenon. The methods contribute as powerful mathematical tools and appear to be further effective, computerized, and user-friendly to investigate nonlinear fractional equations as well as comprehensive analytical solutions of nonlinear evolution equations in engineering, technology, and mathematical physics.

On nonclassical symmetries, Painlevé analysis and singular, periodic and solitary wave solutions of generalized Hirota-Satsuma coupled KdV system

In this work, we investigate the generalized Hirota-Satsuma coupled KdV system which is materialized in the theory of shallow water waves. Nonclassical symmetries of governing system are derived using Bluman and Cole method. The obtained symmetries are in the form of Jacobi elliptic functions, which are more generalized than earlier obtained symmetries for the governing system. Some new singular, periodic and solitary wave solutions of the governing system are also derived by assuming the Jacobi elliptic function type solutions. The obtained exact solutions for the considered system are verified by graphical representation. The Painlevé property of the considered system is also checked with the help of Kruksal’s method and symbolic computational tool Maple which shows the integrable behaviour of the considered system.

Sumudu-Decomposition Method to Solve Generalized Hirota-Satsuma Coupled Kdv System

In this work, we take Adomian Decomposition Method (ADM) and combine it with Sumudu Transform method (STM). This connection between the two methods is called Sumudu-Decomposition Method (SDM), then use it to solve generalized Hirota-Satsuma Coupled kdv (H-SC kdv) systems and also we applied the STM, to find the approximate solutions of system one. Then we compare the approximate solutions of the two way with exact solitary solutions. Clarifying the best way through tables and drawings, then discussing the reason for the changes taking place in the roads and which one is closest to the exact solution.

A unique computational investigation of the exact traveling wave solutions for the fractional-order Kaup-Boussinesq and generalized Hirota Satsuma coupled KdV systems arising from water waves and interaction of long waves

A novel technique, named auxiliary equation method, is applied in this research work for obtaining new traveling wave solutions for two interesting proposed systems: the Kaup-Boussinesq system and generalized Hirota-Satsuma coupled KdV system with beta time fractional derivative. Our solutions were obtained using MAPLE software. This technique shows a great potential to be applied in solving various nonlinear fractional differential equations arising from mathematical physics and ocean engineering. Since a standard equation has not been used as an auxiliary equation for this technique, different and novel solutions are obtained via this technique.

Chebyshev–Gauss–Lobatto collocation method for variable-order time fractional generalized Hirota–Satsuma coupled KdV system

In this paper, the Chebyshev–Gauss–Lobatto collocation method is developed for studying the variable-order (VO) time fractional model of the generalized Hirota–Satsuma coupled KdV system arising in interaction of long waves. To define this new system, the Atangana–Baleanu fractional operator is implemented. The operational matrix of VO fractional differentiation for the shifted Chebyshev polynomials is extracted and then, a collocation scheme is established to reduce the original VO fractional problem to a system of nonlinear algebraic equations. The validity of the presented method is investigated on two numerical examples.

Exact Traveling Wave Solutions and Bifurcations of the Time-Fractional Differential Equations with Applications

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative.

Shapes and dynamics of dual-mode Hirota–Satsuma coupled KdV equations: Exact traveling wave solutions and analysis

In this paper we communicated with three different ansatze methods including the tanh-expansion method, the rational sine–cosine method and the Kudryashov-expansion method to study solitary wave solutions for a new developed nonlinear equation. We presented new generalized Hirota–Satsuma coupled KdV system of second-order in time t involving phase-velocity, dispersion and nonlinearity parameters. The new system can be defined as a dual-mode model where each involved field-function possess a spreading of dual-waves instead of single-mode wave. Graphical illustrations on the effect of both phase-velocity and dispersion-nonlinearity factors on the spacing of the obtained dual-waves are provided.

Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long waves

In this article, two different methods, namely sub-equation method and residual power series method, have been used to obtain new exact and approximate solutions of the generalized Hirota-Satsuma system of equations, which is a coupled KdV model. The fractional derivative is taken in the conformable sense. Each of the obtained exact solutions were checked by substituting them into the corresponding system with the help of Maple symbolic computation package. The results indicate that both methods are easy to implement, effective and reliable. They are therefore ready to apply for various partial fractional differential equations.

Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2-Expansion Method

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

Lie symmetry analysis, conservation laws and exact solutions of the time-fractional generalized Hirota–Satsuma coupled KdV system

In the present paper, Lie point symmetries of the time-fractional generalized Hirota–Satsuma coupled KdV (HS-cKdV) system based on the Riemann–Liouville derivative are obtained. Using the derived Lie point symmetries, we obtain similarity reductions and conservation laws of the considered system. Finally, some analytic solutions are furnished by means of the invariant subspace method in the Caputo sense.

On the New Solutions of the Conformable Time Fractional Generalized Hirota-Satsuma Coupled KdV System

Abstract In this paper, generalized Hirota Satsuma coupled KdV system is solved with tanh method and q-Homotopy analysis method. New fractional derivative definition called “conformable fractional derivative” used in the solution procedure. Tanh method with conformable derivative firstly introduced in the literature. By the graphics of analytical and approximate solutions, it is shown that, both methods provide an effective and powerful mathematical tool for solving nonlinear PDEs containing conformable fractional derivative.

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.

A generalized Hirota-Satsuma coupled KdV system: Darboux transformations and reductions

A Darboux transformation is constructed for the generalized Hirota-Satsuma coupled KdV system and the result is compared with the recent work of Geng and his collaborators [X. G. Geng et al., Phys. Rev. E 79, 056602 (2009) and X. G. Geng and G. L. He, J. Math. Phys. 51, 033514 (2010)]. It is shown that our Darboux transformation may be applied to three interesting reductions of the general system. In addition, the iteration of this Darboux transformation is worked out, and some solutions to the associated systems are obtained.

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

Abstract In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.

Topological and non-topological soliton solutions to some time-fractional differential equations

This paper investigates, for the first time, the applicability and effectiveness of He’s semi-inverse variational principle method and the ansatz method on systems of nonlinear fractional partial differential equations. He’s semi-inverse variational principle method and the ansatz method are used to construct exact solutions of nonlinear fractional Klein–Gordon equation and generalized Hirota–Satsuma coupled KdV system. These equations have been widely applied in many branches of nonlinear sciences such as nonlinear optics, plasma physics, superconductivity and quantum mechanics. So, finding exact solutions of such equations are very helpful in the theoretical and numerical studies.

Application of the simplest equation method to some time-fractional partial differential equations

In this paper, we establish exact solutions for some time fractional differential equations. The simplest equation method is used to construct the exact solutions of nonlinear fractional Klein–Gordon equation, Generalized Hirota–Satsuma coupled KdV system of time fractional order and nonlinear fractional Sharma–Tasso–Olever equation. The simplest equation method presents a wide applicability to handling nonlinear wave equations.

New Exact Solutions of Some Nonlinear Systems of Partial Differential Equations Using the First Integral Method

The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE), (2 + 1)-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.