Generalized Hirota Satsuma Coupled KdV Equation Research Articles

Data-driven soliton solutions and parameters discovery of the coupled nonlinear wave equations via a deep learning method

In this paper, we extend the physics-informed neural networks (PINNs) to learn data-driven solutions and discover the coefficients of multi-component and high-dimensional coupled nonlinear partial differential equations (cNPDEs). In the forward problems, the data-driven solutions include “bell”-type (dark and bright) and “anti-kink”-type solitons of the generalized Hirota–Satsuma coupled KdV equation are investigated and reconstructed. Moreover, some special shape one-soliton and two-soliton solutions such as “Kink”-shaped, “U”-shaped, “Y”-shaped, and “X”-shaped of a (2+1)-dimensional cNPDE are learned and simulated as well. Particularly, some factors affecting the performance of PINNs are investigated, such as the structure of neural networks, the kinds of activation functions and optimizers, as well as the number of collocation points and iterations. For the inverse problems, the coefficients of the equations are successfully identified and driven by various kinds of solution data. The impact of the number and moment of observations on the model is discussed while the network’s robustness is validated by introducing mensurable initial noises to the training data. Numerical experiments not only show that the propagation direction and dynamical behaviors of the cNPDE can be well uncovered by utilizing the deep learning method, but also exhibit the advantage of finding equations with multiple parameters.

Applications of Sine–Cosine wavelets method for solving the generalized Hirota–Satsuma coupled KdV equation

Applications of Sine–Cosine wavelets method for solving the generalized Hirota–Satsuma coupled KdV equation

Solitary Solutions to the Fractional Generalized Hirota–Satsuma-Coupled KdV Equations

In this paper, the complete discrimination system method is used to the fractional generalized Hirota–Satsuma-coupled KdV equations, which depicts the dispersive long wave widely applied in fluid mechanics. First, by the wave transformation, the fractional generalized Hirota–Satsuma-coupled KdV equation can be transformed into ordinary differential equations. Second, a series of new solutions are constructed by the discrimination system method of the fourth order polynomial, these solutions include kink solitary wave solutions, Jacobian elliptic function double periodic solutions, trigonometric function solution, solitary wave solutions, two-period solutions of elliptic function, and implicit solutions formula solutions. Finally, the comparison between our results and others’ works is also given.

A mixed approximate method to simulate generalized Hirota–Satsuma coupled KdV equations

A mixed approximate method to simulate generalized Hirota–Satsuma coupled KdV equations

Bifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equations

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

Formula omitted]-soliton solutions and long-time asymptotic analysis for a generalized complex Hirota–Satsuma coupled KdV equation

A generalized Hirota–Satsuma coupled KdV equation, which can describe the interactions of two long waves with different dispersion relations, is studied in this paper. Using the Hirota direct method, its bilinear form is firstly obtained, and secondly its N-soliton solutions are found. Finally, the collisions between two solitons are proved to be elastic via performing the asymptotic analysis. Some graphs are given to illustrate that the two-soliton collisions do not change any physical quantities except small phase shifts.

Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method

Abstract In this work, the Kudryashov method is handled to find exact solutions of nonlinear fractional partial differential equations in the sense of the modified Riemann–Liouville derivative as given by Guy Jumarie. Firstly, these fractional equations can be turned into another nonlinear ordinary differential equations by fractional complex transformation. Then, the method is applied to solve the space-time fractional Symmetric Regularized Long Wave equation and the space-time fractional generalized Hirota–Satsuma coupled KdV equation. The obtained solutions include generalized hyperbolic functions solutions.

Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative

ABSTRACTIn this paper, we present the exact solutions obtained for the space–time conformable generalized Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative. The conformable sub-equation method is applied to obtain the solutions; the solutions obtained are compared with the extended tanh-function method for the special case when the fractional order takes the integer order. The analytical solutions show that the conformable sub-equation method is very effective for the conformable-coupled KdV and mKdV equations.

An efficient computational approach for generalized Hirota-Satsuma coupled KdV equations arising in shallow water waves

An efficient computational approach for generalized Hirota-Satsuma coupled KdV equations arising in shallow water waves

Numerical Solutions of Coupled Nonlinear Evolution Equations via El-gendi Nodal Galerkin Method

In this research the solution of coupl ed modified Korteweg -de Vries equation (mKdV) and the generalized Hirota–Satsuma coupled KdV equation by using El-gendi nodal Galerkin (EGG) approaches are presented. El-gendi nodal Galerkin (EGG) (EGG) approaches consist of two approaches, the first is El-gendi Chebyshev nodal Galerkin (ECG) and the second approach is called El-gendi Legendre nodal Galerkin (ELG). In these new approaches spaces of the solution and the weak form to the system are presented. The resulted systems of ODES are solved by the fourth order Runge-Kutta solver. The convergence and the stability of these new methods are analyzed numerically. Numerical results are presented and compared with the results obtained by pseudo-spectral method. Original Research Article

A New Fractional Subequation Method and Its Applications for Space-Time Fractional Partial Differential Equations

A new fractional subequation method is proposed for finding exact solutions for fractional partial differential equations (FPDEs). The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. As applications, abundant exact solutions including solitary wave solutions as well as periodic wave solutions for the space-time fractional generalized Hirota-Satsuma coupled KdV equations are obtained by using this method.

(G′/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics

In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann—Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota—Satsuma coupled KdV equations and the time-fractional fifth-order Sawada—Kotera equation. As a result, some new exact solutions for them are successfully established.

Homotopy Perturbation Method for the Generalized Hirota-Satsuma Coupled KdV Equation

In this paper, we consider the homotopy perturbation method (HPM) to obtain the exact solution of Hirota-Satsuma Coupled KdV equation. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear mathematical problems.

On the Coupling of Auxiliary Parameter, Adomian's Polynomials and Correction Functional

In this paper, we apply He’s variational iteration method (VIM) coupled with an auxiliary parameter and Adomian’s polynomials which proves very effective to control the convergence region of approximate solution. The proposed algorithm is tested on generalized Hirota–Satsuma coupled KdV equation and numerical results explicitly reveal the complete reliability, efficiency and accuracy of the suggested technique. It is observed that the approach may be implemented on other nonlinear models of physical nature.

Comment on “A new method for a generalized Hirota–Satsuma coupled KdV equation [Appl. Math. Comput. 217 (17) (2011) 7117–7125

Recently, in the paper [Manlin Xie, Xuanhao Ding, A new method for a generalized Hirota–Satsuma coupled KdV equation, Appl. Math. Comput. 217 (17) (2011) 7117–7125], the authors have applied the two–dimensional differential transform method to approximate the travelling-wave solutions of generalized Hirota–Satsuma coupled KdV equation subject to two different initial conditions. We checked the second initial condition and related exact solution and found that it does not satisfy the original equation. It seems that a typographical error in the initial condition has caused a mistake in the related exact travelling wave solution. The correct solutions have been studied by DTM and RDTM in Abazari and Abazari (2012) [Reza Abazari, Malek Abazari, Numerical simulation of generalized Hirota–Satsuma coupled KdV equation by RDTM and comparison with DTM, Commun. Nonlin. Sci. Numer. Simulat. 17 (2) (2012) 619–629].

The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics

By introducing a new general ansätz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarieʼs modified Riemann–Liouville derivative. By means of this method, the space–time fractional Whitham–Broer–Kaup and generalized Hirota–Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations.

Numerical simulation of generalized Hirota–Satsuma coupled KdV equation by RDTM and comparison with DTM

In this study, generalized Hirota–Satsuma coupled KdV equation is solved using by two recent semi-analytic methods, differential transform method (DTM) and reduced form of differential transformation method (so called RDTM). The concepts of DTM and RDTM is briefly introduced, and their application for generalized Hirota–Satsuma coupled KdV equation is studied. The results obtained employing DTM and RDTM are compared with together and exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by classic DTM. The numerical results reveal that the RDTM is very effective, convenient and quite accurate to systems of nonlinear equations. It is predicted that the RDTM can be found widely applicable in engineering.

Exact traveling wave solutions for a generalized Hirota–Satsuma coupled KdV equation by Fan sub-equation method

In this Letter, the Fan sub-equation method is used to construct exact solutions of a generalized Hirota–Satsuma coupled KdV equation. Many exact traveling wave solutions are successfully obtained, which contain more general solitary wave solutions and Jacobian elliptic function solutions with double periods. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.

A new method for a generalized Hirota–Satsuma coupled KdV equation

In this paper, differential transform method (DTM), which is one of the approximate methods is implemented for solving the nonlinear Hirota–Satsuma coupled KdV partial differential equation. A variety of initial value system is considered, and the convergence of the method as applied to the Hirota–Satsuma coupled KdV equation is illustrated numerically. The obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient. Numerical examples are illustrated the pertinent features of the proposed algorithm.

EXACT TRAVELING WAVE SOLUTIONS OF THE GENERALIZED HIROTA–SATSUMA COUPLED KdV EQUATION

Exact traveling wave solutions of the generalized Hirota–Satsuma coupled KdV equation are obtained by the generalized Jacobi elliptic function method.