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.

Bell polynomials and superposition wave solutions of Hirota–Satsuma coupled KdV equations

In this article, the bilinear form, the bilinear Bäcklund transformation, Lax pair, the integrability, infinite conservation laws, nonlinear superposition formula and superposition wave solutions of Hirota–Satsuma coupled KdV (HSCKdV) equations are studied, which can help us get more properties, increase the diversity of solutions and get more new phenomena. The bilinear form, the bilinear Bäcklund transformation and Lax pair of HSCKdV equations are constructed by using the Bell polynomials approach. Based on this, HSCKdV system is integrable in the sense of Lax pair. According to the obtained bilinear Bäcklund transformation, infinite conservation laws and nonlinear superposition formula are derived. And some superposition wave solutions of HSCKdV equations can be attained by the nonlinear superposition formula. With the help of the symbolic calculation system Mathematica, the properties of the superposition wave solutions are analyzed. What is more, in addition to many properties obtained, the relationship between the solutions is found via the derived nonlinear superposition formula, and mixed solutions containing different functions and arbitrary functions are constructed. By using the Bell polynomial method and studying the integrability and other properties of HSCKdV equations, as well as their exact solutions, it can be found that solitons maintain their shape unchanged during motion and are not destroyed. Such waves have very important applications as a communication method. Furthermore, it has been observed that the results attained in this work have not been presented before.

N-soliton solutions of Hirota-Satsuma coupled KdV equations with variable coefficients

In this paper, we propose a class of generalized variable coefficient Hirota-Satsuma coupled KdV equations, which take into account the inhomogeneity of medium and boundary conditions, and can be used to describe long-wave interactions with different dispersion relationships. The bilinear equations with variable coefficients are obtained by the Hirota bilinear method, 1,2,3, N-soliton solutions are obtained by the perturbation method. In addition, the effects of variable coefficient functions on 1, 2, and 3-soliton solutions are analyzed by numerical simulation, and conclusions are drawn.

New solutions of the time-fractional Hirota–Satsuma coupled KdV equation by three distinct methods

In this paper, new solutions of the time-fractional Hirota–Satsuma coupled KdV equation model the intercommunication between two long waves that have well-defined dispersion connection received successfully by the unified method, the improved [Formula: see text]-expansion method and the homogeneous balance method. In contrast, these methods are simple and efficient, and can obtain different exact solutions to this equation. By symbolic calculation, polynomial solutions, hyperbolic function solutions, trigonometric function solutions, rational function solutions, etc. are acquired. Furthermore, we plot and analyze some solutions.

Application of the Elzaki iterative method to fractional partial differential equations

In this article, we present an iterative transformation method for solving fractional partial differential equations that combines the Elzaki transform and iterative methods. By this iterative transformation method, numerical solutions in the form of series are obtained. When we apply this method to the fractional linear Klein–Gordon equation, we find that it yields the same results, just like the Homotopy perturbation method. The procedures and results of this method for solving the new generalized fractional Hirota–Satsuma coupled KdV equation are given in the paper.

Soliton solution and asymptotic analysis of the three-component Hirota–Satsuma coupled KdV equation

In this paper, we study a class of Hirota–Satsuma coupled KdV equations that can be used to describe the interaction of two classes of long waves. By using the Hirota bilinear method, the 1, 2, 3-soliton solutions are obtained. On this basis, the asymptotic analysis of soliton solutions proves that the collisions between solitons are elastic, and a set of visual figure is given to illustrate the results.

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

Accurate demonstrating of the interactions of two long waves with different dispersion relations: Generalized Hirota–Satsuma couple KdV equation

In this study, the generalized formula of the Hirota–Satsuma coupled KdV equation derived by Hirota and Satsuma in 1981 [Hirota and Satsuma, Phys. Lett. A 85, 407−408 (1981)] is analytically and semi-analytically investigated. This model is formulated to describe the interaction of two long undulations with diverse dispersion relations; that is why it is also known with a generalized model of the well-known KdV equation. The generalized Kudryashov and Adomian decomposition methods construct novel soliton wave and semi-analytical solutions. These solutions are represented in some distinct graphs to show the waves’ interactions. In addition, the accuracy of solutions is verified by comparing the obtained analytical and semi-analytical solutions that show the matching between them. All solutions are checked by putting them back into the original model through Mathematica 12.

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.

Construction of analytical solution for Hirota–Satsuma coupled KdV equation according to time via new approach: Residual power series

In this work, a modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota–Satsuma coupled KdV equation with time as a series solution. Comparison of the analytical approximate solution with the exact solution concluded that the present method is an important addition for analyzing a system of partial differential equations that have a strong nonlinear term. We also represented graphically and discussed the effect of initial condition parameters and reaction of time on the model.

SOLITARY WAVE SOLUTIONS OF SOME CONFORMABLE TIME-FRACTIONAL COUPLED SYSTEMS VIA AN ANALYTIC APPROACH

The main purpose of this research is to inquire the new solitary wave solution of the coupled time-fractional models to validate the influence and proficiency of the planned variational iteration method (VIM) using conformable derivative definition. Applications to four demanding nonlinear problems like Hirota-Satsuma coupled KdV equations, modified Boussinesq (MB) equation, approximate long wave (ALW) equation and Drinfeld-Sokolov-Wilson (DSW) equation demonstrate the efficiency and the robustness of the method. An analysis of the consequences with effects of relevant parameters and comparison with the exact solution presented with the help of graphs tables and gives the further understanding of numerical results by others. The convergence of the method is illustrated numerical and their physical significance is discussed

Formula omitted]-soliton solutions and dynamic property analysis of a generalized three-component Hirota–Satsuma coupled KdV equation

In this paper, a generalized three-component Hirota–Satsuma coupled KdV equation describing the interactions of two long waves with different dispersion relations, is investigated. Applying Hirota bilinear operator theory, the bilinear form of the proposed model is first obtained, and then its N-soliton solutions are given in explicit forms. Finally, the analysis of the dynamic property shows that the collisions between two solitons are elastic.

Closed-form wave structures of the space-time fractional Hirota–Satsuma coupled KdV equation with nonlinear physical phenomena

AbstractThe present paper applies the variation of(G^{\prime} /G)-expansion method on the space-time fractional Hirota–Satsuma coupled KdV equation with applications in physics. We employ the new approach to receive some closed form wave solutions for any nonlinear fractional ordinary differential equations. First, the fractional derivatives in this research are manifested in terms of Riemann–Liouville derivative. A complex fractional transformation is applied to transform the fractional-order ordinary and partial differential equation into the integer order ordinary differential equation. The reduced equations are then solved by the method. Some novel and more comprehensive solutions of these equations are successfully constructed. Besides, the intended approach is simplistic, conventional, and able to significantly reduce the size of computational work associated with other existing methods.

Two efficient computational technique for fractional nonlinear Hirota–Satsuma coupled KdV equations

PurposeThe purpose of this paper is to apply an efficient hybrid computational numerical technique, namely, q-homotopy analysis Sumudu transform method (q-HASTM) and residual power series method (RPSM) for finding the analytical solution of the non-linear time-fractional Hirota–Satsuma coupled KdV (HS-cKdV) equations.Design/methodology/approachThe proposed technique q-HASTM is the graceful amalgamations of q-homotopy analysis method with Sumudu transform via Caputo fractional derivative, whereas RPSM depend on generalized formula of Taylors series along with residual error function.FindingsTo illustrate and validate the efficiency of the proposed technique, the authors analyzed the projected non-linear coupled equations in terms of fractional order. Moreover, the physical behavior of the attained solution has been captured in terms of plots and by examining the L2 and L∞ error norm for diverse value of fractional order.Originality/valueThe authors implemented two technique, q-HASTM and RPSM to obtain the solution of non-linear time-fractional HS-cKdV equations. The obtained results and comparison between q-HASTM and RPSM, shows that the proposed methods provide the solution of non-linear models in form of a convergent series, without using any restrictive assumption. Also, the proposed algorithm is easy to implement and highly efficient to analyze the behavior of non-linear coupled fractional differential equation arisen in various area of science and engineering.

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.

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.

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.